What do "wonk" as against performer and “Wonking out” mean?

"Wonking out", like "geeking out", means to not just be a wonk, but to do so excessively in a way to deliberately show off how much of a wonk one is, often with a mixture of pride (toward fellow wonks) and embarrassment (toward non-wonks).

He's putting up his Powerpoint presentation as if to say "See how wonky I am? I will actually bore you with pie graphs instead of giving you sound bites.", but his aim is really to impress the people who value pie graphs more.

Paul Ryan is often described (maybe self-described) as a policy wonk (meaning (2) in your question). He likes to talk about the technical details of laws and policies. It's boring to most people, because most people want to be entertained with 30-second sound-bites. Entertainers are performers; wonks are like nerds. They may not dress or otherwise look as funny (the stereotype, that is), but what they talk about is called "geek-speak" because it's very technical and might was well be littered with computer terms as economics terms (Ryan's alleged area of expertise).

Think of the phrase to zone out: "to lose consciousness; to pass out". You can zone out because you're bored or smoking dope or daydreaming about being Spiff the Spaceman in the Calvin and Hobbes comic strip.

So wonking out probably means that he started talking about those enigmatic and inherently boring little points of policy to his audience.

Pass out, zone out, and wonk out. All the same pattern. It could be that the out adds an implication of outer space or having an out of body experience.