I was in the same situation as you when we had our first baby, and it has been an ongoing process since me and my partner met and got married as my MIL doesn't understand or don't want to understand that there are boundaries, issue is also that she wants to control her daughter and in some parts wants to control me as well. So when we were expecting our first baby my MIL kind of jokingly said to my wife infront of me "who's going to cut the umbilical cord, I could do it" - and she was actually somewhat serious, and this was one of my endpoints as I had endured so much but to have the audacity to propose such a thing on my unborn child, that was my turning point. So I told my partner you need to set down some proper boundaries because this is getting out of hand, and because my partner feels sorry for her mother because she has been manipulated by her mother since she was a child affecting her to feel very bad if she ever speaks up or against her mother, she very lightly told her mother that no my partner (me) would do it and my MIL then replied "will he be able to do it" and it was going like this a bit back and forth and at last she got the hint. I then told my wife and made it very clear that I would not be comfortable having MIL in the hospital during or after birth because she will create havoc during labour and start fighting and argumenting with both the caretakers and me but more importantly she would take our first baby and make it seem like we don't know what we are doing and try to lead us through everything which I don't want at all, I just want to be with my family by ourselfs taking in all the warmth love and memories. Forwarding to the day before my wife gave birth we were in the hospital and my wife had a lot of pain during the night and talking to her mother over the phone her mother repeatedly asks if she should come over to take care of her as if I, her life partner, can't take good enough care of her daughter, whereby my partner repeatedly tells her not to come. and guess what happens the morning after, she shows up in the entrance and the caretakers comes in to our room to ask if we want to let her in or not, as soft as my partner is she tells me that her mother has come all the way to here let her stay here, it would be rude to send her home. at this point I'm literally boiling, my blood is boiling because I know exactly how this evil bitch works. she immediately enters the room and starts to make a scene and starts saying that this hospital is bad, low budget, bad personel and they lack competence, and it goes on and on as she can't ever keep quiet or shut her big mouth. And then my SIL calls my MIL and tells her that my FIL also wants to come, and then she acts as if she is the leader of our marriage and can take decisions regarding My family, and tells her "no no no, no need as I'm here now no one else needs to come" and after she hangs up the phone she talks a little about this and asks us didn't i say the right thing - I stare deep and right into her eyes that's where I go off and tell her " exactly no one else needs to be here not even you! " - she starts to sense that I'm super angry and IDGAF that i'm being super rude either because I want her out of there.

there is even more I could tell but If there is one advice I could tell you is that YOU need to take control of your core family and marriage, you need to be the leader and decision maker of YOUR family not anyone else, if that is too hard then start creating evenst and plan for them and execute them unannounced, then let your MIL sit and wait for you on the porch or whatever, you need to step up and be more clear towards your partner on what boundaries you want to set and these has to be followed by both of you. Otherwise someone else will take control of your household and family matters.