H is not being honest with you, you get that, right? It was not a backrub, it was sexual activity that involved him talking her off.
OW was right. If he intends to reconcile with you, they cannot be friends. She is not a friend to your M, she has no place in either of your lives. I'm sure H fails to greet and say goodbye to plenty of people at work. There is no reason for them to interact. He has not yet let go. He is not yet being honest with you.
There is no reason for eggshell walking and putting up with this BS. It is counterproductive. But there is also no reason to grill him or put him under a microscope. The stuff here is pretty darn straightforward. Be direct, report where you are, let him choose what to do with that info.
"H, I don't buy that all that was involved was a nonsexual backrub. I want to feel safe, secure, respected, and loved. Right now I feel like an idiot. You have lied about the nature of your R with OW before coming home and after coming home. I require the truth now, clean slate, no attacks, I just need the truth so I can make my own choices."
"OW was right -- our being a couple doesn't fit with a friendship with her. She did not respect our M, she continues to not respect it, she is not a friend to our M. For me to be with you, that R needs to end. I want to know it is crystal clear to her that it is over. You can get on the phone, call her, I'll listen on the extension. Tell her she has no place in your life, no phone calls, no texts, no emails, no unnecessary pleasantries. Let me be clear -- it is not OK with me for you to continue to make room in your life for her. It does not work for me. It is really up to you if you want to keep her on the back burner while you figure out your life. I can see why it would be hard and painful to really let go. It is your choice. But my choice is to not continue to work on a romantic R with her still in the picture as a friend. I will not settle for a crappy R. I want a vibrant, passionate, fantastic, productive partnership with a hot sex life. That means us first, without question, without the disrespect of keeping on friendly terms with your fantasy R."
"Hey, I need the passwords to blah blah blah. Let me get them from you now."
Set some boundaries. Don't settle for a crappy R, you'll just be right back here.