I prefer a Jazz Bass for overall tonal range for recording rock and ease of playing with a faster neck for short fingers.
I also love the meaty sound of a P-Bass even though the neck is a bit of a stretch for me.
I can dial in my Jazz to cover a fat P-Bass sound, but a P-Bass doesn't have the available top end cut of a Jazz.
I also have a PJ and usually run it mostly of the P-Bass pickups.
When I'm tracking my Jazz Bass, I usually have tone all the way counter clockwise using mostly the neck pickup tones.
I also have a Hofner 501 for those specialty tones.
I know temptation may have you leaning towards an all purpose, multi-function bass, but for long term ownership and value retention over the years, there is still nothing as strong as a conventional pre-CBS vintage spec Jazz Bass or P-Bass with passive vintage mild pickups.
You will hear a gazillion opinions otherwise usually genre specific, so concentrate on your own application.
You will hear folks fiercely defending their own personal purchasing decisions because that's simple human nature.
For recording, tried and true through generations of all kinds of music, a passive Jazz or P-Bass is the way to go.
Ask the engineers, producers as well as the artists.
Playing live, I may run my jazz wide open, but that top end treble bite is buffered by my amp and 8X10 AlNiCo cab.
Take your time, play as many basses as you can get your hands on and keep saving so you aren't forced to pass on a great instrument. If you don't have the entire purchase price covered, use layaway.