Consider this example:
I'm sorry if you got the impression that I meant to insult you. That was not my intention.
Would it be correct to say that the above person apologized?
All the dictionaries I have checked defined "to apologize" as admitting one's fault. However, in the above example, the person is not conceding he is making a mistake; he is merely clarifying his intent and, well, apologizing if he hurt the other person's feelings. Yet, based on my understanding of the verb, it would be adequate to say that the person apologized.
So, does apologizing really require being at fault?