I'd rather think of myself as a musician that happens to play bass than a bassist. I listen to it all, melodies, drum parts, chord progressions and whatever else. Denoting yourself as a bassist sort of boxes you in IMO, people assume that you only listen to basscentric stuff.
Drummers the same, I've worked with technicians and I've worked with musicians.. The musicians are looking at the whole picture. Not just concentrating on their bits...