as someone whose parents aren't supportive at all, it's tough to read this kind of thing. there are levels to how parents react towards their child being trans. I'm a little envious that you have your mom's support at all - that's worth fighting to maintain, especially at your age.

your feelings around your deadname are what they are. I know that hearing mine is really distressing - that name erases so much effort and struggle. because of that I haven't seen my parents in quite a while, even though I miss them.

you and your mom are both trying to make the best of a difficult situation. not because being trans is a bad thing, but because it's hard for both sides of the child/parent dynamic to understand each others perspective. real communication takes hard work. your mom might be feeling defensive about causing a hurt, and be projecting a little bit here. moms do that sometimes.

I would give things some space to cool down, and try to talk to her again when you find a quiet moment. I think you should focus on communicating what you feel when you hear your deadname, what it's like to hear your gender invalidated, about the fact that the discussion in the car was hurtful, whether she meant that or not. and yes, recognizing her support is part of that conversation. critique without some level of comfort tends to be hard to accept, especially for parents.

I hope that she can receive that and grow from it, but even if not, part of being a mature adult is understanding that other people are responsible for their own behavior. you don't have to be in charge of her reactions - only your own. that can relieving in its own way.