If you’re looking at Fender, I’d encourage you to get a precision bass over jazz for those styles or if you want a bit more tonal range a PJ (precision neck pickup and jazz bridge pickup).
How much do you want to spend? Are you open to buying second hand?
Willing to spend as much as necessary for whatever constitutes a "pro" instrument (imo a pro instrument is one that has gold standard tone and won't get in the way of virtuoso techniques). Obv this varies from instrument to instrument, like for the string family that's around $3.5k, guitar around $1.5k, and other stuff like banjo $5k or god forbid woodwinds, bassoon at $10-20k.
Fender not a requirement, just want relatively understated styling.
Ok if you want a usable rock tone that cuts through the mix and aren’t looking to slap or get a trebly quack type sound (e.g Jaco) then I think a precision bass is what you’re after. But go play some basses before buying something. Feel and comfort are important too. The jazz bass typically has a thinner neck profile so it’s easier to move around the frets.
Fender has kind of started to make precision basses with J style necks.
One model I fell in love with but never bought was the American elite p bass. A few years ago they replaced it with the American Ultra series which is considered its successor and I think is the same concept but maybe with different pickups and hardware but . It’s a PJ arrangement with a jazz style neck. It also has active electronics meaning it has an onboard EQ that can be switched on allowing you to shape your tone to fit your style. With it switched off it’s a passive bass with volume and tone and pickup selector. Meaning you can turn it into a standard passive p bass as well.
I really think this is the best bass fender produced and combines the best of the p and j basses without sacrificing what makes both great. And it’s relatively affordable on the secondary market for between $1,200 - 1,500 depending on condition. This is definitely a pro level instrument.