there are feminist voices on both sides of the debate, so identifying as feminist doesn't tell me that much about what people believe in that context, only about why they believe it.

And not even that, neccesarily, more the context in which that belief is held.

But yes, I agree with you re: the public perception of feminism. Of course it doesn't help that, as with any movement, some feminists want to imagine that there really is one unified movement, and so either gloss over differences or exclude anyone too divergent from their definition.