DC1 was being very obnoxious towards me and present family, but in that jokey way that can be laughed off to a degree. She didn’t stop when I asked her to.
OK, so your DP told a young adult to stop being obnoxious to her mother and other family members?
I'm assuming that your DD was making insulting/sarcastic remarks, perhaps in a jokey way, but designed to hurt people? This suggests that she walks all over you and is possibly a bully. Is this the case?
I agree with @WhyWouldSomeoneDoThat that family therapy with your DD's would be a good step. They shouldn't be rude to you and other family members and you need to talk this through.
Sadly, I do have a few friends IRL (I'm 51) whose young adult children are rude and unkind to them, despite their parents bending over backwards for them. In fact, it tends to be the most amenable parents who get treated in this way, because their children know that they can get away with it with no consequences.
Given your update, I think this is less about your DP and more about about how your DD's view you and possibly a lack of respect for you.
Edit to say that if either of our children (21 and 17) are rude to us, the other parent does tell them to stop doing it. But it's easier when it's a parent, of course.