Whether it is "begging" or not, the traditional definition of forgiveness is absurd.
Not sure what the "traditional" definition is? I see each one of us as an individual with individual needs and therefore with a slightly different definition of forgiveness. What I need is not what someone else needs.
We forgive ourselves, not others.
This is true, but we can only forgive ourselves and let it go when we know the one we love is not harmed by what we have done or said. Otherwise, it's acceptance of what we have done and moving on with a wound that never can heal.
So why should I hold that power over someone by accepting their apology? Why should I play into and promote that logic?
If what your wife needs to feel better is not important enough for you to hear her out and not dismiss her feelings, I don't understand how she can feel anything, but disrespected and belittled.
I help free her from her mental prison if she feels that way.
Only if she has not hurt your feelings. I don't know anyone who is perfect and has never had their feelings hurt. Sure, sometimes we do it and don't mean it. In that case, maybe she doesn't need to apologize, technically. If you don't listen to her, you invalidate her feelings. In turn, you take a chance of hurting her feelings.
It isn't controlling in any manner to do so. I don't say "I reject your apology". I say that she has no need to apologize. I just want to enjoy every moment. She knows this concept as well, except when she becomes emotional.
I disagree that it isn't controlling. You aren't allowing her to be who she is and accept that she has feelings. Whether those feelings are justified or not, she has them all the same.