A little bit of both, actually. The nickname Bat-Girl was probably given to Barbara a long time ago on one of her adventures, then struck the writers of the penny dreadfuls as a convenient way to get across "nice vampire lady is cleaning up the streets" while keeping Barbara anonymous.

The fictional Bat-Girl, for instance, does a lot more fornicating than Barbara really does, but the writers know that sex sells. Also, attributing the actions of all the vampires (Barbara, Zinda, and Helena) to one Bat-Girl adds to the mystique.