Just on the issue of bringing things up in front of children. DH and I have a rule to always back each other up in front of the kids and then discuss later. We always provide a united front because I think it's important the children know we are a team. I'm not sure if you and DS's dad have that kind if arrangement but I think you could request it. IMO it is undermining to be questioned in front of the child. It sets an example that the parents authority can be argued with, and that is something I disagree. Discussion is ok but that isn't what it was.

As for calling him baby. I tend to agree that he has asked you to discontinue so if it were me I would try very hard to stop and explain it might take a while because its a habit but you are trying.

Of course you can still tell him 'you'll always be my baby' but that is different to baby as a pet name.