Birth of Birthers
A summary of what happened is that the desperate Clinton campaign was looking for things to attack Obama with in late 2007, and made it their campaign strategy to attack his "otherness" (not being American raised/educated, his foreign beliefs, and so on). They didn't start all of the rumors (some predated the campaign), but the Clinton campaign, supporters and operatives, threw everything that was anti-Obama that they could find at the wall, to see what would stick, and a lot of it did.
The media and Hillary supporters argue that since technically, some of these things existed in obscure forms (in seldom read dark corners of the Internet), that Hillary herself wasn't the creator of it... as if that makes her clean. But that ignores the practical reality that it was the Clinton campaign that pulled the birther rumors out of obscurity, weaponized them, and propagated them into public attention. Without her campaign, the Republicans and other Obama detractors, had much better things to go after Obama on than his birth certificate.
The modus operandi was, Hillary's campaign could popularize them, and then after they took hold, Hillary could get around to denying them in passive-aggressive ways (reverse compliments, "Obama's probably not a Muslim, as far as I know", "he probably doesn't abuse his kids, often", and so on). And the plausible deniability that the Clinton's are known for, gives the partisans and the fake-fact-checkers like Snopes, FactCheck and so on, the ability to distort the truth and imply she's completely innocent, or that those claiming she's guilty are completely false -- when they're more like 90% true, just slightly overstated. We can't prove that she was running her campaign (making the key decisions), but if all those folks were going rogue, then she was incompetent or at least attracted those kinds of dirtbags and liars to represent her. Which is worse?
The Press/Progressives also skip the part where all Obama had to do to end the conspiracy theory (or at least cut the legs out from under it), was get people his long form birth certificate and publish it in a competent way. If he hadn't allowed his close supporters to write exaggerated bios of him being some poor immigrant child from Kenya, and stuff like that, it wouldn't have been as much of a problem. Instead it took 4 years (until April 2011), and pressure from Trump and others, to get the administration to do that. And instead of a native scan that people could analyze, they put it through Adobe Acrobat scan compression that a funky layering compression (and OCR) on scans that made it look doctored, which started up another round of accusations. Then add to that some funky timing with key people dying, and a highly secretive administration (lots of questionable associations he doesn't want people looking into, and no one remembering him before Harvard), and you've got conspiracy smörgåsbord. This wasn't just loony conspiracy theorists, this was some really shady or incompetent stuff, and since the Press was proclaiming him to be the smartest man in the room, people went with shady.
Here's a quick reminder of the timeline and what we know:
- 1991 A booklet from Obama's literary agency (Acton & Dystel) was promoting a later abandoned book by him, "Journeys in Black and White". It contained a brief biography of Obama as, "Barack Obama, the first African-American president of the Harvard Law Review, was born in Kenya and raised in Indonesia and Hawaii..." Later, the editor at the time (now partner) claims that it was her "mistake". But that doesn't change that this was fertilizer for believing that if Obama wasn't Kenyan born, he was using the Elizabeth Warren'esque fake indian / poor immigrant schtick when it suited him. And that created evidence for some to latch on to.
- 2004 Obama was accused of being a Muslim and lying about his background, just because many facts in his books did not actually check, out and were confirmed as fake/exaggerated or doctored. Not to mention that if you read his books, he's an anti-colonialist that sympathizes more with his fathers anti-Americanism and surrogate "like a father" (Frank Marshal Davis's revolutionary marxism) than with American values. Some use that as the seeds of the birther movement, and thus claim Hillary is innocent for popularizing it, since it predated her campaign. But it was her campaign that brought them to the mainstream publics attention, before then they were obscure things that nobody knew of or cared about. 
- In 2007 as the race was tightening, a desperate Clinton campaign was throwing everything they had at this uppity Jr. Senator who was threatening their appointment to the throne, and the Clinton machine got caught doing many dirty tricks. There's a little plausible deniability for Clinton herself, but none for her campaign.
- January 2007, Insight Magazine, reported that Clinton's team had questions about Obama's Muslim background. It said that researchers connected to Clinton were spreading the allegation that he spent four years in a madrassa. The Clinton campaign said this was not true.
- March 19, 2007: Penn Strategy Memo: Hillary's campaign strategist circulated a memorandum that proposed exploiting Obama’s "lack of American roots" as his soft spot and what the campaign should target. The goal was to subliminally impugn him without getting caught saying it directly. But as they got more desperate in 2008 it moved to more direct accusations. (We've known of this memo since 08/2008: any news site that fails to mention this in their origins of Birther stories is intentionally deceptive). 
- Dec 2007, Bill Shaheen, a close friend of the Clintons and a senior member of Clinton's staff, had to resign over raising Obama's admitted use of marijuana and cocaine as a youth. Not direct birtherism, though he had floated "unknown background" as well, but it shows that they were getting caught playing dirty. 
- Dec 2007, Judy Rose (usually called a minor campaign worker that they often can't even remember the name of) is forced to resign after forwarding an email suggesting Obama was a secret Muslim who intended to destroy America from the inside. Rose was the volunteer chair of the Clinton campaign in Jones County, Iowa (not a minor functionary) -- she sent the internal email to just a handful of Democrats, and was only fired after the media discovered it. A second Clinton staffer resigned just a few days later for the same offense. If the Press hadn't gotten caught, no one believes she would have been fired. 
- Dec 2007, according to Patti Solis Doyle (Hillary's campaign manager), admitted that a Clinton campaign staffer had, in fact, circulated the Birther conspiracy theory this early. (No one did enough follow-up to figure out which campaign person did this). 
- Dec 12, 2007 heading to the Iowa debate, Obama and Hillary had an exchange where Obama confronted her camp had been sending out crap labeling him as a Muslim. Hillary exploded, but it was clear that Barack Obama's campaign thought Hillary was the start of these dirty tricks: that he was a Muslim and not American enough.
- 2008 editor James Asher (Washington, D.C. bureau chief for McClatchy) remembers Hillary's ally and advisor Sidney Blumenthal was shopping the birther story with him.
- Sean Hannity mentions back in 2007 and 2008, Hillary's campaigns was calling him saying how appreciative they were of him exposing Obama's radical associations. Donald Trump also said they were pushing the same he's a muslim meme on him back them. I'm not sure we should completely trust the word of partisans, but it does seem to support what others were saying at the time too.
- NBC News’ Mark Murray admitted that they shopped the story with him. He tries to excuse her, because nobody ran it verbatim. Here's his tweet, "If someone shopped the story -- and no media outlet ran with it -- is that really "starting" it?". Just because they didn't run with it at the time, doesn't mean they didn't eventually run with it, and he and many admit they started investigating the angle because of the Clinton campaign allegations.
- Of course to Hillary supporters, everyone that remembers things differently than a Clinton is lying. All of them. Which means if Hillary isn't a pathological liar, she has a virus that makes most people who come in contact with her, into one. 
- Feb 2008 Clinton Campaign was desperate. Hillary's Campaign circulates picture to the media of Obama in African Tribal Dress and was quoted as saying, "Wouldn't we be seeing this on the cover of every magazine if it were [Clinton]?" The Obama campaign said this was, “[This is] the most shameful, offensive fear-mongering we’ve seen from either party in this election” and it was, "feeding into false claims on US websites that he is a Muslim". Clinton at first claims she didn't know who did it, then had to let that person go as a "rogue paid staffer" (one of many, before and after). 
- March 2008 - Obama admits he thinks this was a coordinated smear campaign against him from early in the election. (Since the Republican's didn't fear him at that point, it wasn't them). There was a pretty strong implication of who he meant. In response, Hillary apologized and made the claim, "as far as I know", Obama is not a Muslim -- and this started a backlash that she was putting wiggle-room in her statement. It was obvious that Steve Kroft didn't realize, or was soft touching Hillary, that the Muslim rumor was being forwarded by the Clinton campaign at the time she was mildly almost denying he was, in a way that fueled the story. Either she was incompetent, or conniving.
- In Sept 2015 Trump was asked a question. An audience member implied Obama is not an American and a Muslim, and asked about Islamic Training Camps in the U.S. Trump responded to the training camp stuff with, "we'll be looking into a lot of things", and the Press and Hillary campaign tried to make a deal about Trump not attacking a supporter for claiming Obama is a Muslim. Hillary not only did the same thing (ignore parts of toxic questions she doesn't like), but it was her campaign that had seeded or at least fertilized this whole idea in the first place, making her attacks that much more hypocritical. The press omitted that context when reporting on Trump's response.
- Summer 2008 Hillary's PUMA PAC (Party Unity My Ass -- referring to Never-Obama faction of the DNC) was posting updates on the hunt for Obama's birth certificate. Later PUMA got shut down for campaign/501c violations, conveniently eliminating that paper trail. And we don't know if Hillary campaign was guiding them or not, though quite a few in this came from her camp and campaign. But again, only Republicans are responsible for what their supporters do, never Democrats. 
- April 2008 - An anonymous email circulated by supporters of Mrs Clinton, thrust a new allegation into the national spotlight — that Obama had not even been born in Hawaii.
- April 2008 - the Obama campaign responded by posting a copy of his “certificate of live birth” — a shorter version of the birth certificate. Which is accepted as proof of birth from applicants for a US passport. But it was not the full, original certificate — and that the campaign failed to show the serial number and other details in their scan. Also there were legitimate questions about Hawaii's standards at the time (they did issued short forms after the fact, to kids born abroad). And then there was multiple Social Security numbers associated with Obama, issued from a state he never resided in, and it was all feeding on the seeds of doubt that the Clinton campaign had sowed and fertilized. Which meant that the conspiracy theorists were fueled by the incompetence/ambiguity and secrecy of Obama, as well as questions raised by the Clinton campaign. A poorly scanned document only fueled them more.
- April 2011 Trump pressured Obama to finally release his long form birth certificate to try to end the speculation (after letting it fester for 4 years). However, it too was scanned in a particular incompetent way to provide proof. Most would have used a raw image (tiff, or scan format). Instead the administration used Adobe Acrobat, which does a weird planar (layered) compression, that happened to recognize differences between different inks or fields in ways that fueled the conspiracists much more than it should have. Had the administration handled any of it competently, it might not have eliminated everyone who had doubts, but it would have dramatically reduced their numbers.
- April 2011 Politico Ben Smith and Byron Tau explained Clinton supporters circulated an anonymous email questioning Obama’s citizenship, (Barack Obama’s mother was living in Kenya, Obama was born there and his mother then took him to Hawaii to register his birth thing) -- while this is technically the "birther" meme that started it, everything before that was part of it too, "He's not American enough, is a Muslim, was educated in a madrassa", and so on, were all the seeds and soil that the birther claim took root in. And the Clinton campaign was involved in all of it. Yet the apologists will claim, "she didn't start it". Yeah, right.
Fact checkers show their bias
Some absurd "fact checks" conclude the following degrees of falsehood:
- Hillary's campaign didn't do what they did
- Well, they did it... but it didn't count because some of the rumors existed before they brought it to the public attention.
- Well, maybe they popularized it, but who cares because Republicans are evil.
- And besides, it was just her campaign staffers that did it (following her top strategists plan), but that's not her personally.
- She might not have known what her staffers and strategist were doing, so everyone that says otherwise is making it all up.
It's only Republicans that should be held accountable for what their followers do, not Democrats.
All of these are embarrassing for the "fact checkers" credibility. If they wanted to be the slightest bit non-partisan, it would have been at least half true, and they would have explained that while Hilary or her campaign wasn't the only origin of the stories, they were gleefully spreading them and responsible for popularizing them from the very beginning. While we can't prove it was under Hillary's orders, if it wasn't, then she wasn't in control of her own campaign machine. The fact checkers might also correct the record and add new information as it came to light, but they prefer pointing to stale mistruths and dismissals, rather than admitting to facts that don't reflect well on their research. So readers are left missing a ton of context (called lies of omission).
Compare their versions to the historical record above: