What jumped out at me as I read this article from Contemporary Psychoanalysis was how many versions of non-apology I would get in one interaction with either my sister or my mother. In the process of one "apology" they would use every one of the iterations of non-apology defined in this article. I had to get into my forties before I could start to recognize that these women had never apologized to me for anything in all my life. I went through this recently with my sister. She had pulled one of her stunts and I called her on it in no uncertain terms. She waited for a couple of weeks and then sent me an email with her "apology". Oh, it was a grand attempt. What she didn't anticipate was how transparent her explaining, blame-shifting, appeals to her perfect intentions, etc. were to me. I nailed her between the eyes with my rejection of her "apology" and told her we have no relationship as far as I'm concerned. I'm sure she choked real good on that one.
I suspect that you've experienced this version of a non-apology from the narcissist in your life, too. The next vile incarnation of a non-apology is "Explaining". I now quote:
A related substitute for apologizing is the practice of explaining. Unless the listener is particularly sensitive, an explanation can sound remarkably like an apology.The lesson of the day is: Don't confuse an explanation for an apology. Even if the person started with what sounds like contrition, if they follow it up with an explanation which lessens their culpability in some way it is the same as putting a "but" between two sentences. The "but" negates everything that preceded it. Every single time.