Ah, if only I could do the same as you describe. Unfortunately I've been accused too often of delivering a fauxpology, with particular scorn for anything she perceives as "I'm sorry you feel that way". I do think there's a difference between "I'm sorry you feel that way" vs "I'm sorry for hurting your feelings [even though I don't think there's anything wrong with what I actually did]", but I think that difference is too subtle for her. Or maybe she just really wants a guilt-laden apology (she has said this at times).
I think maybe this comes down to the perceived moral dimension to these (claimed) transgressions. There are mistakes like bumping into someone by accident, and then there's punching someone in the face. In both cases we may apologise, but in the first there's hopefully no moral condemnation by the victim nor is there much of a feeling of guilt by the bumper. If both parties accept that the case in question is more like a bump than a punch, maybe it's feasible to give a "sorry for my role in your feelings being hurt" kind of apology. But if she sees it as a moral failing ("how could you be so inconsiderate?!?" etc) then, in my experience, nothing short of "I'm so sorry, I know I'm not considerate enough, I'll work harder to do better, please give me another chance" will do.