Your cheating
Did your cheating violate any commitment you made to your H2b? IMO, no commitment, no foul.
If you did violate a commitment, what did you do to change into someone who won't violate a commitment again?
Also, he probably had a lot of feelings when he found out and may have a lot of unresolved feelings now. It's up to him to resolve them. A good IC can help.
Your M
Your past M isn't all that relevant now, although one or both of you may decide not to R on the basis of it.
Can/will you share why you became the partner your described?
His cheating
He did not cheat because of a lousy M. He cheated because of his own issues. Discussing his grievances is a waste of time, unless it's a path to his taking responsibility for his actions and himself.
To be a candidate for R, he needs to acknowledge to himself and to you what he's done/doing, he's got to see that he's terribly flawed as a partner, and he's got to commit himself to changing from cheater to good partner.
You report he's not doing those things. He's not yet a candidate for R.
Your healing
You've got to process the feelings that are getting dumped on you by his continuing to stay in contact with his ap. That usually takes a long time, because usually there's a lot of emotion.
You also need to figure out what you want, even if you think it's unattainable. If you want R, you need to own that, even though it looks like a long-shot. If you want D, you need to own that, even if you stay together for, say, financial reasons.
Above all, you need to be honest with yourself. You need to think straight and talk straight. If anyone on any forum or if any self-styled expert tells you to act in a certain way, because that will manipulate your H into acting the way you want him to, ignore ignore ignore.
People are not machines. Even if your H 'always' does this when you do that, he may surprise you; besides the rules have changed. Ask for what you want - any attempt to manipulate will backfire eventually.
And the idea that anyone who doesn't know your H knows how he'll respond is totally ridiculous - some predictions will come true, of course, but that's coincidence.
IOW, no one knows how your H will respond if you force a choice between her and you.
I recommend not forcing a choice until/unless you're ready for him to choose her (and if he does, he's not worth your time anyway).
I also recommend reading the Healing Library; a link is in the yellow box in the upper left of SI pages.