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.
Think of it this way. Your wife doesn't want an apology.
For instance, I can say "I'm sorry for that lousy thing that happened to you last weekend."
I have no idea what you did last weekend, but I'm sure one lousy thing happened. So in a way, sure, I'm validating your feelings over something I don't know anything about...which probably feels really hollow to you. How can Pook apologize for something he had nothing to do with and doesn't even know about?
That's how your wife feels because she doesn't want some random "sorry" thrown her way. It's just words and it's meaningless.
What she actually wants...but what she never says...is for you to understand how she feels.
And you may think, who the heck knows how she feels. She screams and then cries and then picks a fight and then wants to go to Dairy Queen like nothing ever happened.
I get that because I've been there. We all have. It's maddening and makes no sense at all.
However, you know what it feels like to be angry...or sad...or depressed...or frustrated. If you saw that in a kid, or your mom, or really anyone, what could you say or do to help them process through those feelings? You'd talk it out, you'd comfort them, you'd help them calm down in the moment and get a grip on what's happening.
That's what your wife wants you to do...even though she'll never actually say that in a million years. What she wants is for you to be loving and kind and tell her that everything will be okay.
Then why the screaming and all the drama, you ask? Well, that's the mental illness part of this...it doesn't make sense. It's overflowing, out of control emotions that can't see left from right, and it causes pure panic in every possible direction. Your job is to talk her past that, let her know you're not going anywhere, and just love her until the moment passes.
Maybe you'd say, "But you don't get it...it's ALWAYS LIKE THIS!!!" I do get it though. Once you start to connect and she can calm down, then it's not these crazy blowouts anymore. All she's looking for is love and support...even though she says a billion other things.
What she really means to say (but never says) is that she's insecure and has no idea what to do about it, or how to handle it, because you'd never understand. And maybe you can't. But you can understand sad or mad or other emotions...and you know how to comfort and soothe them. That's where you start and it makes all the difference in the world.