Of course Fender doesn't make it any less confusing with the Jazz Bass Special model, which is a P body, P/J pickups, and a Jazz neck. It's a Precision Special in every other variation except this one. Is it a Jazz bass? Fender wrote that on the headstock. But it's not a Jazz body by any means.
Jazz neck and Jazz bridge pickup + Precision body and Precision neck pickup make it at least equal parts J & P.
To be more confusing, the passive Precision Special of a few years back has the same basic set-up.
FWIW, I also have a Warmoth with J neck, J pickups, and P body. It sounds and plays like a Jazz to me, except it has a smaller, lighter body. (think G&L JB-2, etc.)
I think of the Fender Hoppus bass (P neck, P pickup, and J body) more as a Precision than a Jazz, despite the name.
Maybe we should call the hybrids with P and J pickups PJ if they have P body, and JP if they have a J body...
I also remember the Fender JP-90 bass that had J neck and body with PJ pickups.