The neck is usually what determines which Fender bass you play. Usually the defining measurement on the Jazz Bass is the 1½" neck at the nut. The Precision Bass is normally 1.625" wide at the nut. Unfortunately, although the width has stayed the same, - the thickness of the necks have varied a lot over the years. My 1965 Fender Jazz that I bought in 1966 had the shallow C shape and the 7.25" radius on the fingerboard. That was a great neck and very comfortable and fast to play on. In 1966 Fender put block markers on the neck, which looked really cool, but when I pulled it down to try it out I was shocked to find it had a deep U-shaped, very thick neck.
After touring using my '65 in the early 70's my '65 Jazz was stolen in 1976. It took me almost 10-years to find another Jazz with the same neck specs. My newer one (an '82 Jazz that I bought in 1985) is a maple neck/fretboard, but has the same dimensions and radius as my '65. Pretty much all modern Fender Jazz basses, even the 60's re-issue, use a slightly flatter 9.5" radius. If you can find a Jazz with the 1963 shallow C neck, I think you'd like that. Doubt if you can get a 7.25" radius on it, but you'll find that shallow C neck to be very comfortable.
I never liked the neck on a Precision Bass because it was not only wider at the nut, but just plain chunkier all the way around. However, my grandson got a Squier Precision Bass kit (includes small amp) for Christmas and I was showing him a few things on it. I was surprised that the Squier P-bass he had was much shallower than any P-bass neck I've ever used and I was fine with using it. He'd been using a loaner Fender P-bass from his instructor and the neck on that one was much thicker than his new Squier.
So, yeah, if you like shallow, thin, necks, start trying out the various Fender Jazz basses out there. Just be aware that you will find some with some pretty thick necks, so be sure you try them out before you buy them. If you order it online, make sure you have a grace period to return it.