Does accepting the one require giving the other?

To genuinely and sincerely accept an apology I would think some amount of forgiveness is required. If you're being genuine and sincere.

That echoes my interpretation of forgiveness...unfortunately it also means I tend not to forgive people so much as acknowledge their apology. My acceptance ends up being insincere. (Then again, most of my grudges have to do with unrepentantly thoughtless and discourteous people who offer insincere apologies for their transgressions.)

Most of my forgiveness comes independently of an apology. If someone recognizes they've done me wrong and doesn't do it again, that's enough. And for people who habitually step over lines without changing their behavior, all the sincere apologies in the world won't soften my disposition towards them.