This is a Jazzmaster:

Original flavor. No disputes.
This is a Jazzmaster: (American Original)

Different fretboard radius. Is that all?
This is a Jazzmaster: (Troy Van Leeuwen)

Slightly altered rhythm circuit controls.
This is a Jazzmaster: (American Professional)

No rhythm circuit, pickup selector moved.
This is a Jazzmaster: (Player)

No rhythm circuit. Different pickups / humbuckers.
This is a Jazzmaster: (American Performer)

No rhythm circuit. Stratocaster style bridge.
This is a Jazzmaster: (Jim Root)

Strat style hardtail bridge. Active humbuckers. Controls limited to volume and blade pickup selector.
This is a Jazzmaster: (Blacktop)

No rhythm circuit. Humbuckers (there is a version with a humbucker in the bridge, Jazzmaster pickup in the neck)
THIS is an Offset Telecaster:

THIS is a Parallel Universe Jazz Telecaster:

(Sorry, can't find a full body image)
If the Offset Telecaster is a Telecaster, the body shape does not define a Jazzmaster.
If different pickups can be used, the pickups don't define the Jazzmaster. (Even though they are called Jazzmaster pickups?)
If different controls can be used, the controls don't define the Jazzmaster.
If a different bridge / tremolo system can be used, that doesn't define the Jazzmaster.
So what does that leave us?
I submit, for your consideration, that a Jazzmaster is defined, according to Fender, by the headstock shape. It is the only constant across everything Fender defines as a Jazzmaster. The headstock. That would be rather anticlimactic, right?