To me, there are two levels to the answer to this one.
One level is what other musicians will consider a Jazz Bass when you turn up at the audition. For this one, you have to understand that most non-bass players are not up to speed with the various basses out there. For them, it must say "Fender" and preferably "Jazz Bass" on the head stock. My '62 has the black headstock; so, Fender left off the "Jazz Bass" part, and that seems to be ok.
Among us bass players, things are quite more wide open. But I tend to get a little confused about any strict definition. My Tobias has the 2 pups and an offset body; but, I would not consider it a Jazz Bass. I would call my Sadowsky a Jazz bass. So, I don't know where any hard line is. It probably varies a bit from player to player. I respect whatever each cat wants to think, as I'm certain each has their own reasons.
Now, get back to practicing....