Ok, which one & why?

Depends. I'm a jazz bass player myself (have a custom bass made by Chris Stambaugh, a luthier who lives in NH. Here she is:



I started off with a Fender American Deluxe Jazz; it sounds fancy, but it's just bells & whistles really, a Mexican J-bass is plenty. The jazz bass has a generally rounder, bass-y tone, with the option of using just the bridge pickup & playing further towards to bridge to get that biting Jaco-like tone.

The Stingray has a more metallic sound, and uses a single humbumber instead of two single-coil pickups, so it's a much hotter, buzzier bass. But, it's really great for rock-style slap (Flea's main bass was a Stingray until Modulus endorsed him & built him a Stingray clone.) Best bet? Go play each one, see which you like. And always remember that you can have the action adjusted on them. The one thing about Fender's J-basses is that they have a big neck. Great sound, but a bit tough to play sometimes. My custom bass (which was NOT expensive) has a thinner neck, which makes it easier to play. Just a thought. Good luck!