I generally make it my priority to do two things when I play:
1. Keep perfect timing - I will not exceed a certain level of complexity if I feel it will compromise my ability to stay 'super-glued' to the beat of the tune. However, of course when I am practicing I push the limits or else I would have never gotten past playing quarter notes at 120 bpm.
2. Integrate the Harmony - I will also sacrifice speed and flair if I feel that it will detract from expressing the deeper layers of a tune's chord progressions. For example, if I am playing a tune at 140 bpm and the chords change twice a measure AND those chords happen to be other than just three-note triads (say, 7th chords or add9's), I will go slower to express what is 'really going on' with the chord progressions to the listener.
Other than that, I tend to play progressive rock and jazz. For jazz, I play in a band called The Lancaster Jazz Project which plays in the post-bop acoustic style but with electric instruments and a rock drum kit.
I like to stay in 'the pocket' but I also enjoy tastefully inserting quirky bass-runs when I can. I try to avoid just hammering on and off from root to octave or from root to fifth to octave UNLESS the song calls for it and I am making it rhythmically 'interesting' kind of like some of Jamerson's best (of course, I am not saying he played only root-fifth-octave, I mean that some of his alleged more 'simple' songs are more complex than meets the ear because of the rhythmic complexity).
I like the bass to be pretty quiet in the mix and it almost ALWAYS surprises people I play with or record with because they assume I will want to make the bass really loud in the mix. I prefer it to be as quiet as it can be without detracting from the sound.
I like a strong low-mids, weak high-mids, and strong treble tone, preferably with a setup that uses one single coil pickup and one hum-cancelling pickup. I like things like P/J basses for instance as well as Twin Humbuckers that can be switched into a position so the neck pickup is single coil.
I hope that helps elaborate 'what kind of player' I am!