When there is any kind of conflict, she doesn't apologize off-the-cuff, in the moment. That kind of apology tends to have the goal of quickly defusing things so everyone can just move on with life already, sheesh. It can be useful, but is rarely satisfying.
She also does not under-apologize. This is the kind of apology that politicians are known for making in rushed press conferences. "I'm sorry if you..." The under-apology doesn't actually take responsibility for anything. In fact, more often than not, it shifts responsibility to the other party. You say you were hurt, silly you, so I guess I'm supposed to apologize now. There. All better?
Neither does my daughter over-apologize. This is the sort of apology that is, consciously or not, designed to elicit some backtracking, qualifying, almost-apologizing from the wounded party. I am such an awful person, I'll just stop speaking entirely so I don't hurt you again. Here, the wounded party has a choice. Does he/she respond with "Okay," which makes him or her a heartless wretch? Or the more desired, No, no, you aren't awful. Please don't stop speaking. I'm sorry I over-reacted. See how that works? Clever, isn't it?
No, my daughter instead walks away from the heated conversation (which used to infuriate me, until I learned to understand her process - now I appreciate it more than I can express). She isn't walking away in order to dismiss the situation, or ignore it. Instead, her goal is to understand what just happened--most specifically, her part in it. Once she sorts it all out and gets her heart settled, she returns. Even if the wounded party is still hurt, my daughter walks right into the tension and says, with direct eye contact and a sincere heart, "I am sorry." No qualification (underapology) or manufactured exaggeration (overapology). Then she goes on to explain exactly where she went wrong -- not as an excuse, but rather as a specific way to describe what she understands she needs to do differently next time.
It is truly a beautiful thing to hear and to receive. The best part is the softening that occurs in my own heart when I'm the receiving party. It melts my bitterness and defensiveness, and lo! I suddenly see clearly how I had reacted poorly, and am able to articulate it and apologize in return.
This is how to build bridges and foundations and create rich, authentic relationship.
It is a wondrous thing, to learn from your own children...