Maybe it's just me, but most p-basses I've played sound muddy and blah, but a few that I've heard on YouTube have quite a defined and aggressive low-mid growl. I really don't think it's a matter of technique either because I dig in and play close to the bridge and I can get a good growl from most J and humbucker basses, but not P basses.
Is it because the good sounding p-basses are being played through tube amps? Are there different types of P pickups, like how Js have single and dual coils? Is it pickup placement?
I think what you are describing really has nothing to do with technique, and everything to do with primarily the pickup itself.
Perhaps the neck and body wood add it's own "color" but the basic footprint, sonically is established by the pickup.
Recently I obtained a 2011 Fender American Standard P5 and installed a Bartolini 58CBP pickup.
The stock pickup had far too much upper midrange and lack of focus in the lows for my tastes.
This change alone brought me about 80% closer to the tone you described which is, punchy, focused in the lower body of the voice, with a natural extension of ugh frequencies. All the EQ'ing and attenuation of highs that I had to do with the stock pickup was no longer needed with the Bart.
My feeling about the Precision Bass pickup location, and overall design is that by it's nature, has a big sonic footprint.
To make it "punchy" the EQ changes are in the direction of less low end in the 30-60hz range, a slight increase at 120-150hz, and again around 800hz.
The tone I still have in my mind that I was chasing was a result of the equipment I had back in the 70's, Alder/Rosewood P-Bass, GK head, small reflex Sunn cabinet with a D-140. The speaker cabinet wasn't even capable of deep bass, and I always loved the punch and definition I got from it. No tweeter either, perfect.
Granted my present Fender P5 has a maple fretboard, so thats going to flavor the tone somewhat, but without question the Bart pickup gave me the essential tone "right out of the box" I was looking for.