Getting My Ex Back (We have kids together)

Hello Everyone.

So I want to summarize the back story for everyone. My ex(31) and I (30)were together for 8 years. He broke up with me days before our 8 year anniversary mark. We have three kids between us. 1 we have together (3 yr old), and then we each have one from previous relationships that were short lived.
He broke up with me April 10th. So it hasn’t been that long but it feels like an eternity to me. He also started to talking to someone new less than two weeks after. I am not sure if they are still talking. The day we broke up he actually did it through text message while he was at work. He broke up with me because he found in my phone that someone was flirting with me through a game platform and I did not stop them initially. At first, I told him I didn’t see the issue since it was a game. That was my mistake. I have since owned up to it and apologized for hurting him and killing his trust in me. I also wrote him an email a little over a week after the break up just outlining how I felt, apologized, and things in the relationship that caused me to get to the point I was at. At the end I said that I want to remain cordial for our kids and also that I am always here for him and I’d like us to rebuild our friendship and if it turns into more I am open to that as well.
After that email we had a phone conversation and throughout the entire conversation I could barely get a word in without him bringing up any little thing I did that he considered a mistake in the relationship over the course of 8 years (which was only four things according to him).
So after that I implemented a modified no contact. I only spoke to him if it had to do with any of our kids. We have not spoken about us since April and that phone conversation. I forgot to mention that we were still on the same phone plan but a week ago I cancelled my line on his account and started my own so that we can have some degree of separation. He did not ask me to do this, I did it on my own. We had a period of 10 days where there was no contact completely not even about the kids. Just complete silence. This ended 3 days ago. He contacted me first and asked how I was doing and how the kids are doing. My son also had a school event coming up. So he and I saw each other face to face at the school event yesterday. My parents were there and so were his. Everything was very cordial and friendly.
I did get a vibe that maybe he might want to spend some time together but I’m not sure. Today I texted him Good Morning with a smiley face and also asked how our son is doing because he went home with him after the school event. He texted back Good Morning with an exclamation point and let me know he was fine. I then told him I appreciated him for coming to the concert and he responded with a No problem. That was the end of the conversation. It wasn’t negative but it was neutral.
I’m not sure if my situation is doomed or what to do from here? I am also not sure if I will see him when it is time for our son to come back to me because he may have his parents bring him back if he’s at work.
Should I go back to the modified NC or should I wait for him to contact me and hope it is not just about the kids? At this point I think I need something that isn’t about the kids in order to break the cycle? Please help :confused:

Okay, you’ve told him everything there is to say about the guy that was flirting with you and apologized for hurting him and losing his trust. It will take a while for him to get past your “infidelity” and the ball is in his court as to whether he wants to reunite. Give it time and continue no contact except as it pertains to the kids. And don’t initiate contact! He will eventually start to miss you and the relationship you had before the other guy issue…