OP, give him time (also give yourself time here please)! As a pp said he’s likely had way less time than you to get used to the idea, plus he is 13 which is a tricky age anyway and his dad is emotionally abusive which likely creates some confusing feelings for him about this whole situation.
Was your DS aware that this change in living circumstances was about to happen or did he just come back from holiday to find you living elsewhere?
Definitely don’t push him to do overnights or give him an ultimatum, your mum gave you some truly TERRIBLE advice there. You will definitely only make things worse that way and push him away. It’s great that he was able to spend happy relaxed time with you in the daytime, you can definitely build on that relationship as time goes on. I think it’s highly likely he’ll come around to overnights before too long, though it may take a bit of time, but as a pp said, the most important thing here thinking long-term is preserving a good relationship with your son, and not how many overnights he ends up having with you. You want him to be calling you on your birthday in 20 years time etc! Play the long game and have patience. A child is going to want to stay where he feels comfortable and relaxed and not pressured or guilted to act a certain way or do certain things. Your ex is right that at this age in practice, no court is going to force a 13 year old to stay somewhere they don’t want to for whatever reason.
I know it’s hard. Try not to take it so personally, it’s likely much more about continuing to stay where he sees as home than deliberately choosing his dad over you. Do make sure you tell him very clearly what you’ve said here, that you left his dad, you did not leave him and you’ll always be his mum and he will always have a place in your home.
I understand how awful this feels, I also left an abusive husband and it absolutely floored me with grief anytime my son spent overnights and weekends and holidays with his dad. So I get it. And I do understand the dramatic language and indeed feelings from someone coming out of an abusive relationship. But although there is some grief for what once was, it REALLY isn’t the same as your child dying (and yes there will be people on here who sadly have had a child die which is of course a billion times worse than this situation, so it’s worth being sensitive to that). I had to forcibly keep reminding myself of that, that however painful it is for me to be living apart from him even some of the time, at the end of the day my child is healthy and alive and happy and that is the most important thing of all.
Finally I wanted to say that you should be really really proud of yourself for getting out of an abusive relationship. PLEASE stop second guessing yourself, you do not really want to go back to a man who threatened to urinate on your things and thought that was a reasonable thing to say do you?! There is no having a good marriage and happy family with a man like that sadly. Grieve for the loss of what you thought you could have had at one point, that happy intact family, but understand that the reality is that sadly with this man that was never on offer as an option. And it is likely you’ll end up with a much better relationship with your son because of splitting up ultimately, he will grow up understanding that you have respect for yourself and that bad things his dad might say about you are not true or fair, as he’ll be able to see for himself who you are away from all that. DO NOT GO BACK!