But what about the fact that apparently some/many jazz pianists don't really improvise very much in their performances and play their songs pretty much the same each time. Seems like the line between jazz and classical, if one says that improvisation is the the key element of jazz, is somewhat blurry.
I assume that most everyone would classify Keith Jarrett as a jazz pianist(except when he's playing classical)but are his renditions of, for example, Over The Rainbow different enough to be called improvisation?
When a classical pianist performs a note for note transcription of a solo by Tatum or Waller does it become a classical piece because the pianist is not improvising?
What about when a classical pianist like Gabriel Montero plays one of her improvisations using mostly classical harmonies and classical figurations...is that jazz or classical or some hybrid?
Great points, but I still don't think you're seeing the difference between improvising and "jazz." I know there's no real definition of jazz, but I'd say it's commonly accepted to use certain harmonies and sensibilities. It swings and much of the early material was built from popular songs and their chord progressions. The approach is very different.
Gabriela Montero is improvising, but it's not "jazz." She is not using the influence of early blues, showtunes, or swinging in a certain way that signifies to a listener that this is "jazz." Again, it's a very hard genre to define, but if you start from when it started, you see what it is and what it isn't, and also where it's at now. Montero is absolutely amazing at improvising, but when she does, it's most of the time an improvisation "in the style of" or "on a theme of."
Whereas, most modern jazz, even today (which i listen to heavily), the formula is basically the same: A head (melody), solos on the harmonic material by each soloist, the out head. That is a very different approach than someone like Montero takes, or even Jarrett. Jarrett doesn't think of himself as a jazz musician, even though he's played in that setting. Much of his improvising is just "music," like he says.
I'm not saying there are rules to playing jazz, but there are certain signifiers that let you know the lineage of where it came from. You can listen to Coleman Hawkins and then to a modern artist like Orrin Evans, or Jeremy Pelt. The formula is mostly the same, and the swinging is there, albeit more modern. The harmonies are there, but more modern.
Also, I would never pay to hear someone play an Art Tatum transcription, to me it defeats the purpose of what Tatum was trying to do. And yes, even though his versions have similarities, it's mostly because he liked what he came up with. Around that framework, things are constantly changing, however small the details. Sure, the intros of Tea for Two might be virtually the same, but many other things change.
You're posing some great questions, and as we all know, how do you define classical or jazz? I'm just putting forth what I believe, and I would say many musicians that have played both and listen to both might say as well.
To summarize, there is a difference to me between improvising and playing jazz.