You said this:
"It's just a deal-breaker to me. It was before and it still is."
Then you said this"
"if I go that route, that probably just means we are heading for divorce then. I am still not sure about it. "
Based on your two quotes above, it isn't a deal breaker for you, at least not anymore. If it was, you'd be divorcing him.
Now, that's OK, it's your life and your choice but you are saying one thing and doing another is my point.
Since you aren't sure about divorce (and that's OK), that means it isn't a deal breaker for you anymore, if it was, you'd be 100% sure and divorcing him already.
Your lying cheating husband isn't showing himself worthy of reconciling, at least not yet, not even close.
Make demands. He WANTS to stay with you.
Tell him if that's what he wants, then he will do ABC and XYZ. If he doesn't, he's choosing divorce as you won't even consider taking him back until things are worked through.
Counseling is a must, if he won't, he's telling you that he doesn't really want you back.
Go on and on with your demands.
If you don't want him coming around, tell him. I get it's his house too, but if you say it, mean it. Either he moves out or you do.
Don't tell him he can't come over and then let him come over. Again, he can come and he can live there, legally but you can choose to move out if he insists on being there. I'd tell you to talk about that with an attorney BEFORE you move out though.
Just don't say one thing and do another.
Set, maintain and enforce proper boundaries.
I mean, he knows he messed up but even with that being the case he still isn't really working on things. If you take him back under these circumstances, he's going to feel bullet proof.
He needs to win you over, you don't need to win him over.
Be strong, don't say one thing and then do another. If you say it, mean it. The old "say what you mean and mean what you say" is important for you in all of this.
Don't let him walk all over you. He's the one who is the lying cheating spouse and he's the one doing little to nothing to make you feel better.
You can't make him do things, work on things. All you can do, if he won't, is to leave him.
I strongly suggest you talk to a lawyer about things, knowledge is power and you need to have more arrows in your quiver, in case. It's better to be prepared, ahead of time.
I'm not saying you have to have him served, but at least begin talking to an attorney.
I mean, your husband is doing pretty much nothing in terms of actually trying to reconcile with you so based upon that, you need to be talking to an attorney as that's where your husband is leading this to.