The main part of the question asks whether other kinds of music have formal frameworks that in any way effect what's played.

No. Lots of arguments about "what is jazz?", "what is early music?", or "what is country?", but these arguments, as far as I can tell, don't have any effect on what actually gets played. Nor do arguments about "what is folk?" have any effect on the playing of folk music. The only very tangential connection might be that a specific club or festival organizer might well book mostly music from one side or the other of such an argument, but it's not because of the argument. It's because that's the music the organizer likes and/or thinks the audience will like. These debates really don't have any effect on the actual playing of music.