Some families make a whole thing and include everyone and that is great. But I would say the majority do not.
I cannot believe they sat you down and put you in that position. They should have talked to DH if that was their request.
You need to be 100% comfortable with everything going into labor. It was really odd for them to bring this up now and get you upset so close to birth, making me think they would not be very helpful in the room.
I have no idea why FIL would honestly want to be there. If DH felt he wanted his mother or parents there, then why in the world did he wait this long to pull this card.
I understand it is a double standard but in this case I think if you want your mom there that is okay, it's toally different and I know he doesn't get that.
I asked my mom to be there in the room with DS1 and she said no, ahahha. She said it was a private moment between me and DH. My sister was there for like an hour before she got bored and left, ahah. They were kicking in the door once he was born, but the labor part they understood it was my decision and special to DH and I.
Good luck dealing with that situation but you need to get on the same page with DH first, then let him handle his parents. As far as your mom goes, I would give him full power to kick her out if he feels she is over stepping her boundaries in the delivery room. I think that is fair that she is there for support for you, but that he gets to be in charge of helping bring his child into the world.
For DS2 it was just the two of us and it was great : )