Am I the only one who's guessed that the plot twist ending has everything to do with Haley having actually been abused by the father and not by this Mr. Klaus person?
Here's why I don't think so:
1) As you note, Haley has a good relationship with her father. With Klaus she's tried to take the blame herself, tried to ignore the issue, gotten angry at her brother for bringing it up, she was very clearly disturbed when she met Other Haley, she's still rattled whenever her brother tries to talk about it, she was never going to contact a therapist herself, and she wasted an entire therapy session by being Haley at her most evasive. Whereas she's obviously quite happy, even eager, to spend time with her father.
Haley's calm and balanced around people she doesn't feel threatened by: her brother, Lisa, Sandra, Diane, Victor, Simon, and her father. The only other time we've seen her act like this was when Adriana was being so hard on her; even though she was disgusted by Adriana's husband she actually planned to seduce him just to get back at her boss, which is an obviously insane idea.
That's a pattern of behavior. Klaus clearly rattles Haley, and I doubt it's because she hated math.
2) How her mother could "know" about Klaus (though we don't yet know if she does, or if so what she knows) and not do something about it remains an issue to be addressed. But it's not very difficult to justify: Haley begging and pleading with her mother to keep it secret, because the alternative —
everyone knowing what happened — in what looks to be a fairly small town would've been doubling down on her trauma. Now, in the real world I'd of course say that something should've been done anyway...if for no other reason than to stop him from doing it to someone else...but this is fiction and the author gets to write the history. In any case, this is another delicate storytelling beat and they'll have to work hard to justify the decision.
Whereas I absolutely
can't see their mother, at least as she's currently characterized, staying with a man who abused her daughter. She certainly wouldn't countenance Haley cuddling up to him on the sofa while watching soccer. That would make her an accomplice to...even an enthusiastic supporter of...the abuse, and that's an insane leap of logic vs. what we've been shown. Even if it was just about some nutty concept of competitiveness she'd be trying to keep Haley and her father apart. She doesn't. Not once.
So no, I don't think there's any chance that it's her father who's the true abuser.
Finally, "Daddy" is a D/s thing. Bog-standard terminology. That her brother doesn't get it is mostly because he doesn't crave it like Haley does. The best he's been able to do so far is use it for a little bit of play. Haley wants something much more intense, but unless he makes an earnest attempt it's probably not going to happen.
I think people conflate Haley's submissive brat persona and her desire to be dominated with the fact that she also has a complicated relationship with father figures. If her Mom's been treating her this way for a long while (whether or not Klaus was involved) and thus her only sources of emotional support were her father and her brother, that's part of it. Reread her early fantasies about her brother and it's clear that back then, even though they were the same age, she envisioned him in a protective and paternal role. Add Klaus to that mix and it's no surprise that she has "daddy issues," but that's not what's making her call her brother "Daddy" when she's trying to induce him to chain her to the bed and do naughty things.