I think this categorisation is a bit of a fool's errand. Are we trying to somehow get past postmodernism here? Or are you re-organising your record collection Woebot?
Seems to me new genres emerge from cross-pollination rather than as direct lineages.
Classical music was incorporating folk themes from well before even modernism, then onto incorporating jazz and that's just an early example, I'm sure there are earlier and for sure there are many many more.
I think zhao's "argument" (you seem to like the quotation marks zhao) is dodgy, not because there's nothing to it, but rather because it establishes the idea of an unbroken lineage, which to my mind suggests that there are two kinds of music - the 'right' kind from the historical tradition and the 'wrong' kind, leading to a questionable, not to mention rather dull, purist approach. Of course the real dichotomy is between 'good' and 'bad' music which is up to the listener. Also yesterday I dropped a cigarette in the cat bowl and banged my head going up the stairs, both by accident, so it's not true that there are no accidents either.
On top of ambient as throwing a spanner, what about harsh noise? It seems to fit into the category of classical music except it's from the other end of the social scale (almost said 'wrong' end lol) and it's made for other purposes. Is that jazz too then? Or rock in its aesthetic? Or it could be considered folk, except it's pretty electrified. Seems only a minute ago everything the Wire touched was ambient, now it's all jazz.
These categorisations are fun but only productive when they start to break down, and any is as good and bad as any other. Like massrock said we're just having fun with the categories and ideas. Very postmodern I know but I can't see a way round it.