Welcome to the forum S. Thanks for the long message. I can imagine this has been hard for you. I have a little empathy having been there, but I am struggling with the fact that you hate the word "affair."
You sound like a spiritual man who is used to bring grounded in that and I can imagine that there is little room for raunchy words such as "affair" and "cheating." They are words rooted in the lower chakras I think and perhaps this is something to consider. It sounds like you feel comfy in the higher chakras, but unfortunately or fortunately we are made of all.
The fact that you had an affair and cheated means that you too have all the the chakras within you. I think you should embrace that and be a bit more humble about it, a bit more of a mirror to yourself.
Look at your wifes face, you fucked up big time in your endevours with this woman. Use her as a mirror to yourself. See the pain you caused. You say you are doing that, but something makes me think that you aren't entirely. Or you would of cried. Long and hard. Crying for your wife. Not you.
I bet that the sex wasn't fantastic and raunchy and rooted in the most grounded chakra. Admit that, be humble about it, its more becoming I think to be humble and ashamed in moments like this. To give of yourself entirely to that without ego.
Please consider that what you DID is as important as the spiritual aspect. Its two fold. You had actions, based on desire and connection (spirit), that is nothing to be ashamed of, but needs to be admited in terms of saying out loud (the two need to be whole), yelling it if you have to "I fucked up! I had an affair, I cheated on my wife!" perhaps then you will cry; when you realize your integrity is now damaged and broken and she has potential to never trust you the way she once did. Somewhere in her is a piece of hatred and resentment I think. That will never be shaken I don't think.
I hope your meeting went well. I would think you had some glee in seeing your two loves together, but I would be very careful with that. This is not your moment. This is your wife's. Everything should be given to her. She deserves to be bowed down to, head lowered, in shame and apology in order to rise up again.
This would be very hard to hear and I don't expect a response that is positive. I would ask that you realize I am not unlike you. I understand entirely what it means to love more than one. To cheat in order to fulfill that and what it means to feel that spirit that is connection, but I have also realized that this is not my journey alone and I need to be realistic, humble, respectful, full of integrity and empathy and not think with ego ever again if I am to have my hearts desire that I need so much.
I hope you take the time to read the threads tagged with "cheating" "affairs" etc. Do a search and get comfy with the word and all it means. You have entered that realm and that is only a detriment if you allow it to be. You can change everthing for the better if you focus on the process to do so. The journey has just begun, I know that. At this time you can embrace it and take it on, one day to educate others and help them with their journey, having learned and become a better man than you ever thought possible. I'm excited for you
you have a lot to do, but you seem like the type to do it!