Definitions are there to draw a boarder. Marketing brands are there to makr profit from associations and emotions.
So before asking what defines a Jazzmaster, it's maybe worth considering what we expect from such definition. If we want to know what to look for when looking this up in a Fender catalogue, it's sure worth clustering and modelling Fender's marketing strategy, even if they start calling a Warlock with JM Pickups and Gibson headstock a JM...
(They will not, obviously, for copyright reasons, even if marketing were inclined to do so...).
However, this might not be the primary goal of why musicians buy a Jazzmaster (I dare not say guitar players do not buy for name mojo but only for musical or technical reasons - but some actually prefer to have some guidance as to the latter ... And I think it's safe to assume that most discussions like that are controversial because they discuss two non-related aspects under the same polarizing thread title...).
So I'm not going to discuss Fenders marketing strategy - If the result has not been nullified by history it will be by future... From a musical-technical point of view, I define a guitar by tone (options) and playability, with certain personal options. Playability includes the body shape, the neck profile and the frets - which are, obviously, a matter of personal preference and often subject to variants (e.g. thinline, CBS vs. pre-CBS etc.).
The typical sound of a JM is IMHO created by scale length, the bridge/vibrato type, the pickups, and their circuit. Again, variants may apply (vibrato position, p90 in JM covers, no rhythm circuit, etc.).
So when does a guitar variant start to be a different model, technically? For me, the typical tone of scale length, specifically developed vibrato and pickups is constitutional to a JM. Plus, I do dig the iconic, slick, and ergonomic offset shape
So when I look for a JM, a thinline, a JMJM, a CIJ, or a pre-2012 VM with hardtail and buckers is not what I am looking for - even if the body shape looks the same in 3D or is "offset" in 2D. The CP trem placement some might prefer, but was a deal killer for me. OTOH, I don't care much for the rhythm circuit others will expect in a JM. All of which may be good on it's own (to some even the "better" JM), but not as a substitute for why I (!) want a JM...
So, aside from marketing, there is a technical core blurring into variants extending to your personal tolerance/preference. What I have learned is: Discussing the use of wording is not helping me in any way...