I distinctly remember Jaime remembering how Tywin diligently helped him overcome his reading difficulties. That's the work of a parent. Perhaps it's one of the few really warm things about Tywin, but there it is.
As for Cersei... she's regarded as overestimating her own abilities by some of her contemporaries. I can't remember if it was Varys or Littlefinger, but I'm inclined to think the former. Basically he/they said that her attractive femininity was her biggest asset rather and her mind. From her own internal monologue she seems to have believed the opposite: her femininity was the only thing holding her back.
Unfortunately for Cersei, she doesn't appear to be as clever as Olenna Tyrell. Not stupid, but not particularly exceptional. So Cersei's resentment of her father, from reading, does look more like a conceit. I do think she went through life looking down on others. Refusing to acknowledge Tyrion's abilities even as she seemed threatened by them (and his position as hand of the King) seems consistent.
But worst of all, Cersei definitely inherited her father's desire/need to dominate and to control. People with such needs aren't likely to feel good about themselves unless they have their worth externally validated through domination, they are - in a sense - weak. I don't think trauma serves as a cause, as plenty of people in worse situations have not manifested domineering tendencies, while people in good situations still do. Tyrion was treated worse than her. Danny, likewise (though I'm aware she may get nastier... not a huge TV fan). John Snow had it worse in some ways than Cersei too.
With that domineering need, Cersei's exercise of power was therefore always personal, rather than professional. That acute need had to be sated regularly, and because it was strong she sometimes satisfied it to the detriment of her own interests (never mind the interests of others). So no, I don't think Tywin traumatised Cersei as we would commonly accept it. She just felt horrible when meeting someone she couldn't control. And with Tywin, she never felt close to doing that.