The basics of any guitar are 1) that it is physically playable (i.e. that it is not a cheese or hard boiled egg slicer posing as a guitar), 2) that it plays in tune (with itself), and 3) it stays in tune. Beyond this it is largely what you do with it.
Is the once budget T-50 beginner guitar with a cheap Micky Mouse pickup a Jazz guitar? Objectively in comparison to high-end offerings from Guild, Gibson, and boutique luthiers, one would have to say definitely not. But what happens when one finds its way into Ted Greene's hands? Hmmmm... suddenly it is a very credible Jazz guitar. Given my choice of listening to Mr. Greene on a T-50 or a whole lot of other players on expensive high-end Jazz guitars, who would I choose? Mr. Greene.
Is a Telecaster a Jazz guitar? It is if it is in the hands of Mr. Greene or Ed Bickert.
Is a Strat a Jazz guitar? There was a guy with the handle, Doc Bop, at JazzGuitar.be who brought some videos to my attention of a guitarist from L.A. playing exquisite straight Jazz on a Strat. Lovely tone. Sadly I cannot remember the player's name now.
And even if the guitar does not meet the three criteria above ...
There was a Quebecois sessional employed for the summer at a college where I taught. He had gigged in Montreal in Latin Jazz bands. He found a garbage Chinese-made plywood nylon string guitar in a storage closet at the college. It had high action. a misplaced bridge, and worn, notched strings. He evaluated its dis-intonation, mapping it in his head, and he then put the strings strategically out of tune. Suddenly, he is playing Baden Powell tunes on it, bending notes into correct intonation on the fly. In his hands it was a Jazz guitar.
I wouldn't say no to the new Jazz box model if someone offered me one. Whether or not I could transform it into a real Jazz guitar, though, is questionable, but I wouldn't mind trying.