Whoo-boy, this old saw?!
I've said it before, and it looks like it's gotta be said again; certainly, people can call their basses whatever they want...Precisions, Jazzes, Tom/Dick/Harry, hunks o' wood and metal what make tunes an' stuff, etc., etc., but the body-shape is the foundation and the starting point. Cut something into an offset, Jazzy shape, it's still a Jazz bass of some sort, regardless of what electronics are packed into it at a later date.
One doesn't have to believe me, just ask someone who cannot use this-or-that body shape because of it's ergonomics. You could cram the most expensive electronics from an Alembic, Kubicki or Wal into a Jazz body, but if a player doesn't feel comfortable with that body's shape, the control suite is completely irrelevant.
And, yes...I keep bringing up Jazz basses because of my personal preference, but it's also to make a point in the form of a
rhetorical question; if the pickups determine an instrument's foundational nomenclature, then which one of these offset bodied, Fender-shaped basses of mine are or are not Jazz basses?