[Nils Jacob Holt Hanssen]

So improvisation, while a key component component of jazz, is not the only or ultimate criterion for assessing whether something is or is not jazz

Agreed. I would say that things like syncopation and the use of certain harmonies are elements that will define something as “jazz”, just as much as improvisation will.

Although, I just read in the Williams biography that he considers improvisation to be the most important thing in jazz – at least from the musicians’ point of view (Williams calls it “freedom”). 🙂