An excellent thread! For me the sound I like comes almost entirely from the bass itself. Pedals to some degree too, but that’s only to provide sounds which would be impossible to achieve with the the bass alone.
I never really found “the tone in my head” until playing a Yamaha BB1025x a couple of years ago. It was grin inducing in a way that I’d never experienced from a bass before. Since then I’ve made sure I have at least one P-style bass in my arsenal.
A recent discovery is that I have a lot of love for MM humbuckers too - they seems to cut through certain mixes better than a P, yet retain a degree of fullness often lacking from soloed Jazz pups. I’ve found my MTD Super 5 (dual coil-switchable MM buckers) to be the perfect Neo-Soul/R&B machine. Smooth and deep yet defined and cuts through a mix surprisingly well.
Once upon a time I might have been in the camp of having “my sound” and sticking with it regardless of the context but now I think of myself as a bit more of a tonal chameleon. I’ll play the bass with the tonal configuration that works best for the music at the time. As has been said, though, the goalposts do shift if you’re in a covers band vs an originals band. I’m not in a band at all, so I suppose I play covers by default!
In summary, I guess I don’t always know the bass sound I want immediately but I do know that I want the bass itself to be the primary factor and that I don’t want the amp to be colouring the sound at all. Those are the only two constants.