Libertarian billionaire David Koch isn't a birther, exactly; he just thinks the president is a Marxist who got all his ideas from his father:
David agreed. “He’s the most radical president we’ve ever had as a nation,” he said, “and has done more damage to the free enterprise system and long-term prosperity than any president we’ve ever had.” David suggested the president’s radicalism was tied to his upbringing. “His father was a hard core economic socialist in Kenya,” he said. “Obama didn’t really interact with his father face-to-face very much, but was apparently from what I read a great admirer of his father’s points of view. So he had sort of antibusiness, anti-free enterprise influences affecting him almost all his life. It just shows you what a person with a silver tongue can achieve.”
So Koch has his facts basically right; Obama didn't know his father very well. But he nevertheless insists that the Democratic Party platform can be boiled down to the "antibusiness, anti-free enterprise influences affecting him almost all his life," which were somehow transmitted by the father he never knew. This piece is unironically titled, "the paranoid style in liberal politics." This isn't straight-up birtherism, exactly; it's an attempt to shoehorn facts into an interpretation of the president as an anti-American radical whose political inclinations have their origins in East Africa rather than Hawaii, Cambridge, or the South Side of Chicago.
This seems like a good opportunity to write a birtherism lexicon to distinguish varieties of birtherism.
Post-birtherism: Any rhetorical construction that acknowledges the president claims he was born in the United States, but leaves open the possibility that he's lying about it. Post-birthers may actually believe the truth, but refuse to make definitive statements in order to avoid offending actual birthers. Examples: John Boehner, Hillary Clinton.
Ironic Post-birtherism: Making humorous or ironic references to the idea that the president was not born in the United States as an attempt to signal solidarity with or otherwise placate those who genuinely believe the president was not born in the United States. Examples: Tim Pawlenty, Rep. Raul Labrador.
Pseudo-birtherism: An umbrella term that encompasses all the various modes of belief that involve embracing fictional elements of the president's background, from the belief that he is a secret Muslim to the idea that he was raised in Kenya. Includes highbrow forms of birtherism like the "Kenyan anti-colonialism" thesis and theories that his name was legally changed to "Barry Soetero," as well as the idea that Obama's "real father" was one of the handful of random black celebrities you can name off the top of your head. Examples: Newt Gingrich, Andrew C. McCarthy.
These aren't "official" terms of course, but this is how I organize the different varieties of birther and birtheresque beliefs, some variety of which appear to be held even by the party's most elite benefactors. I'm sure some people could come up with other/more/better terms. A majority of Republicans already believe in garden-variety birtherism; the number who subscribe to one of its subcategories is probably overwhelming.
UPDATE: Brendan Nyhan suggests* a term for another variety of birtherism, "Birther-curious," which I'm adding to the lexicon. Katya asked for something like this in the comments.
Birther Curious: The belief that, despite widespread availability of the president's birth certificate, that there are "lingering questions" that could be answered by him releasing the "long-form" version, even though Hawaii doesn't issue those. The birther-curious may acknowledge as a rhetorical concession that they "believe" the president was born in the U.S., but nevertheless argue that it's the president's fault that garden-variety birthers continue to exist. Examples: Sarah Palin, Lou Dobbs.
UPDATE II: Based on Dave Weigel's recent article on Donald Trump, I'm adding the term's "Orthodox Birther" and "Reform Birther" to the lexicon.