exactly! It does not sound classical! Try to find a real classic performance of the piece and then hear the difference.http://www.youtube.com/watch?v=BhgcXqWQhkQ
I certainly wouldn't consider that "not-jazz" either.
Just because you're playing something "based" on what a "classical" composer wrote does not make it "classical." I also would definitely not consider that alto sound even a good representation of a "classical" sound. Nice Jazz sound though, therefore..... jazz section.