SCh, FWIW I would step back and detach further. Your H is still spinning like crazy. He is cycling right now and that's why reconciling talk is fading. Just think if you'd taken him back he would probably be on his way out again and you would be starting this all over.
Validating what your H says doesn't mean that you agree with him it just means that he has a right to feel that way.
Do not lie and say that it was mutual a) because it's not true and b) because even if you do it won't fix him. Right now he is still projecting and blaming you for the sitch. You can't help him now he has to look inside and deal with his issues. Until he does this he will continue the rollercoaster ride.
Just continue to GAL and take care of the kids and yourself. It's the best thing you can do for you right now. Leave your H to God.
