Since this thread is about all things New Orleans, I waited for 17 hours for someone else to chime in and give you a thumbs up on Wynton, but it didn't happen; maybe you should put an add in a local newspaper to tell them about this thread.
No one can give a "strict definition" of jazz, and anyone who plays what they have determined is a strict definition of jazz will sound old fashioned, to say the least.
All those names you gave are like "steak and potatoes", it's good, but not everyday; although Mingus is the one exception I would make, that's because he's somebody different everyday.
https://www.youtube.com/watch?v=wdQuL1Vyy04