I believe that he has made an effort to tell you he is sorry and that he wants you back. He seemed to have made that clear...
He told you he was sorry and asked if you wanted to try again. He told you he had lied and wouldn't lie again. He keeps asking to come over. He has told you that he isn't hanging out with the OW anymore. He has asked to go back to Counseling. He did come over and try to fix the furnace. He did follow up to see if it got fixed.
These are all excellent signs of what many on here would give their right arm to hear from their wayward spouse.
I see no reason that you couldn't invite him over to talk. Face to face. Clear the air.
I say to you again, that the purpose of this site is to save relationships. It looks as if you can save yours now. You have shown him you meant business. He needs to know that if it EVER happens again that it is over for good.
I don't think that you need to snoop to find out if he is lying. If you invite him to have a talk, is when you need to gather all the information on his plans and exact relationship with the OW. It would also be the time to see where both of you want to go with this and how you would prevent it from happening again and what would happen if it did.
You need to be careful and you need to be observant to see if his words match his actions. If you decide to try, I would recommend that you tell him you want to take it slow.
You still have the option of continuing on with what you are now doing. I don't think he is going to give up. He seems to be trying to find his way back to you, but none of us know for sure if he is trying to have his cake and eat it too. I would make sure that he tells you face to face that it is over with the OW and that he has ended it with her and will eliminate her from his life for good. I wouldn't tolerate him saying he wanted to still be her friend. That would be the deal breaker and the thing that would cause me to tell him that you aren't interested in gettting back together if he chooses that route. The choice is his.....
I think he wants you back and maybe has learned his lesson. It is still in your control, which is what you want in these situations.
The down side is that you are putting yourself in the position of getting hurt again. I believe that the possibility of getting hurt is on the table no matter who you are in a relationship with. There are never any guarantees that we won't be hurt.
