Jerry R, As you say, jazz is as much a feeling as a series of particular chords. And, most traditional folk music at least IS based on three chords. Valid point, but when musicians start modifying chords, the tune often gets modulated, too. That's one of the most striking things, to me, about "jazz" renditions of folk and pop songs or even of classical music pieces. The tune almost always gets rearranged or used as a springboard for improvisations on the theme.

I'd agree that if you keep the folky style and just fancy up the chords, it's still folk. (I was kinda jesting about "Home On the Range" and "Wild Mountain Thyme" being jazz if you used different chords.)

On the other hand, the feeling/style of a song can also change without changing the chord structure much. I've heard a simple tune such as Gershwin's "Summertime," done as folk, opera (or folk opera), pop, jazz, etc., with little or no modification of the chords, but with changes in tempo, syncopations, vocal style, etc. I don't know what chords Gerswhin specified that song for "Porgy and Bess," but it can sound pretty jazzy without any chords more sophisticated than A minor and E7th. (Of course, if you're playing the sax or trumpet, chords aren't involved, so it's gotta be something else that makes it jazz as opposed to pop or folk.)