And there was the amusing rift between popular, jazz and classical students in terms of what music was about and later working as a session player of sorts I used to laugh (internally) playing jazz sets with some function bands because they didn't play any "actual jazz" and didn't play it "properly" and yet nobody else would even notice.
It's fun to debate music and genre definition, genre lacking enough nuance to even work effectively for lots of music.
If I recall Soweto Kinch (UK rapper and sax player) had difficulties with record stores putting him in jazz, and ideally he'd wanted to be in urban sections also.
For musicians and composers I think genre definition is more of a necessary evil than something helpful.
It's particularly noticable where you have a long running series that allow for the motifs to mature over time.
It's actually about lineage. Contemporary classical sells itself on the basis that there's a direct academic - sometimes historical - link to Beethoven, who was the prototypical Genius Composer.
Which is why you usually get all the stock tropes - concert halls, serious audiences in seats listening very intently, serious clothes for the players, a serious conductor at the front, clapping at the end only, definitely no cheering or dancing - that also apply to traditional classical music concerts.
It's also about intellectual content. Pop doesn't usually try to justify itself intellectually. Contemporary classical does nothing but. Which is why it's a long series of "isms" of various kinds, all making more or less convincing claims about their own immense cultural importance.
You don't get points for writing interesting music, you get points for creating a new "ism" - preferably an "important" one - and then writing music around it.
The committed audience is actually tiny - five figures globally, if that - so "important" is debatable.
Even so - the idea that contemporary classical is "important", with the implication that other music isn't, is probably the key to the genre.