You're mixing up terms here. Gravy seals is an insult term, used for typically out of shape people who are heavy into gun/militia/i-am-very-badass type culture.

The latter two are just ribbing jokes about the Air Force, from the other branches usually. My old (Army) boss used to tell me to 'take off your air force gloves' if he ever saw me with my hands in my pockets.

Not having served (but I did help out on Operation Code for a couple years):

"Chair Force" does sound like something they would say about armchair generals. No?

Absolutely. Just pointing out that one is strictly a pejorative, while the others would likely be viewed as 'someone in the airforce.' I think OP was wanting things more of the former, like gravy seals, meal team six, y'all queda, etc.

I like y'all queda!