I think it's like this. You stumble, smack into someone, and you apologize. Your apology is for hitting them, not for meaning to hit them. In fact, you did not mean to hit them, you just stumbled.
But when the issue is offense, you don't want to apologize, because you think that they think that you meant to cause offense, and apologizing will reinforce in their mind the idea that you really did mean to do it, or at least in your mind it will reinforce that idea in their mind. And you don't want to reinforce that idea in their mind, partly because it makes you look bad, and partly because the idea is simply wrong. So you don't want to apologize.
(Note well: "you" here means "me, too, given the right circumstances".)
Worse: That may really be going on in the other person's mind. They may really think (or at least suspect) that you meant to cause offense, and therefore are determined that you must apologize as penance or proof of repentance or some such. "The good of society demands that I force this person to apologize!"
So maybe the best answer is to say something like: "I did not intend or wish to give offense. Apparently I did; I apologize for doing so." Because the fact is, the offense was real. Apologizing for it is just good manners. But apologize without agreeing that you had intent.
This presumes that you didn't have intent. If you intended to offend, that's a whole different situation. It's worthy of an apology, not just for the offense, but for the intent.
But when the issue is offense, you don't want to apologize, because you think that they think that you meant to cause offense, and apologizing will reinforce in their mind the idea that you really did mean to do it, or at least in your mind it will reinforce that idea in their mind. And you don't want to reinforce that idea in their mind, partly because it makes you look bad, and partly because the idea is simply wrong. So you don't want to apologize.
(Note well: "you" here means "me, too, given the right circumstances".)
Worse: That may really be going on in the other person's mind. They may really think (or at least suspect) that you meant to cause offense, and therefore are determined that you must apologize as penance or proof of repentance or some such. "The good of society demands that I force this person to apologize!"
So maybe the best answer is to say something like: "I did not intend or wish to give offense. Apparently I did; I apologize for doing so." Because the fact is, the offense was real. Apologizing for it is just good manners. But apologize without agreeing that you had intent.
This presumes that you didn't have intent. If you intended to offend, that's a whole different situation. It's worthy of an apology, not just for the offense, but for the intent.