I hope you're in counseling. And I hope you read this in the spirit in which it is intended: to help jolt you into taking action to get yourself out of infidelity and on toward your new and wonderful life.
Because I'm going to tell you how other people might view what you're doing. And the part that's important for you to understand is, YOU don't get to control how other people view what you're doing. You are constructing all of your behaviors and words to create an atmosphere of your choosing, to cause other people's thoughts to be such and such. And I'm here to tell you, from painful experience, that you are setting yourself up for more pain down the road.
Let's review the facts. This man is a lying, cheating, immoral prick. No one can deny that fact. Fact #2 - for some reason, you love a lying, cheating immoral prick. (If I knew the reason for this, I'd have saved my own life much sooner so don't argue, just accept the fact). Fact #3, this lying, cheating, immoral prick has decided he doesn't love you anymore and is doing everything he can to get away from you in some regards but also manipulating you to let him have his cake and eat it too with the kids and coming to the house whenever he likes. And, this is just my opinion, but I believe that Fact #4 is that you hope that by being nice and "forcing" him to do certain things that he'll start to love you again.
So the most important fact here is, you're not working on yourself, trying to sort out your own pain and anguish, heal from the trauma. Instead, you're spending all your time plotting and planning ways to force him into still being part of YOUR life. Whether it's physically or emotionally or in the eyes of others.
I'm very afraid this is going to end very badly for you if you don't change this trajectory. You've received very good advice on this thread and you also got very good advice on your previous ones. That advice hasn't changed nor will it because it all comes from personal experience and knowledge learned the hard way. The other fact for you is: the people giving you advice (or let's phrase it as "ideas for healing" since I don't like the word advice in this situation) are sharing their own experiences. We're not reading your story and saying "oh, my, that's an unusual situation, let me think about ways I can give her ideas to help herself and her children." Hell no. What happens is, a whole bunch of your 70,000+ friends read your post and IMMEDIATELY know how you're feeling and know for a fact what will help get you to the other side with a minimum of pain and agony. We're not just pulling this stuff out of thin air; we know because we lived it and we've watched it happen over and over again. These things follow a pattern and these situations all have more in common than not in common. Your situation is no more special that anyone's. It might be a little different from other in some ways but it's not special at all; sadly, it's the same old story of cheater, cheater, liar, liar.
In closing, I will make one comment. I can't believe you bought that jackass a gift to celebrate his new apartment, to celebrate that he left you and the children, to celebrate his new wonderful life that will be so much happier without you in it. I can guarantee you he might someday use your gift to tell a new conquest he's trying to win over how crazy is ex-wife was and that he had to escape it to survive and that, in the end, you agreed with him so much that you even bought him a gift. THAT is how he will twist your kind gesture.
He is not a good person. He's playing with you. He doesn't care at all what happens to you. He has left you. He has moved on without you. He doesn't want to live with you.
Your children can see all that with their own eyes and you are not fooling them by pretending to be all nicey, nice. They're probably hating the togetherness because they know it's fake.
Please, please, please, read the words of advice that have been offered to you and really try to sit and absorb their message. Either that or let him move back in and consider yourself in an open marriage cuz that's essentially what you're announcing to the world. I know that's not the message you think you're sending but that's the message people are hearing.