I don't really care what the general public is used to. I mean, it matters to DC, of course, but not for my preferences. I feel she has been demoted from someone who debuted as independent hero, a peer, to sidekick (or rather, former side kick) in many regards. I don't like it. Even if everyone else loves it, I still don't like it. I don't care if she's a librarian at all. I do care that she comes on the scene as an independent hero, and that she never be an underling/sidekick/protege of Batman's. I do care that her dynamic and relationship to Dick be more important than the one to Bruce. I much prefer her running the BoP (preferably not out of building Bruce owns or with Bruce's equipment) than to running Batman's electronic security/IT, since that makes her seem like an employee to me. And it seems like "make work" since he manged his own just fine before she had the job. As Batgirl, I prefer her independent. It matters to me.
And yes, her relationship to Jim is most important, but I don't see why it needs to be mentioned in this regard, which about Bruce and Dick. Unless you're comparing father/mentor figures, which I really dislike Batman in that role for her.