I think it's a big thing that a lot of aparents go through, especially in the beginning. Right now you are a new mom and still "claiming" that motherhood and all the feelings that go with it. It can bristle when others don't get it even if they aren't meaning to hurt.
What I would suggest is putting the names back on them when it's used.
For example..
"How is dd's mom?" "I'm great, thanks for asking!" "You know I meant her bmom!" "Yes, but I call her first name and would appreciate you using the same name. Later when dd is older you won't want to confuse her by asking her how her mom and dad are because she's going to think it's me and dh, and rightly so". "She'll know xx by this name and I would appreciate it if you would support that".
Sharing a title comes later, imo. Right now it's yours and that doesn't negate the importance of dd's bmom at all. Rather, it cements yours.
This is just my opinion, but I think over time, the title doesn't bristle as much and it doesn't seem to be as big of a deal as it was in the beginning. At least that is how it was for me.
