IMO, my Fenders are Precisions, and anything else that is basicly a copy can be called P-bass, or P style, and I have no issues with that. Just don't call them a Presicion.
Yes, the Single Coil P's fall under the same rules, but I typicaly referer to those as Single coil P's (or Precisions) to avoid confusion with the much more common, split coil P. When someone says Precision, or P bass, my default setting is the 57 and later split coil version.
Some would go so far as to refer to other basses with the P p/u configuration , but different body styles as P's. I draw the line there. If they don't look like a Precision, and that would include having a split coil Pickup, then it's not a P bass IMO.