The teen years are very, very difficult for most parents and kids. But you and your mom have an unusually serious communication problem. It looks to me like the two of you are resorting to ways of “fighting” that are hurting you both. She is resorting to dismissing your concerns and threats. You are resorting to an eating disorder and blaming her for it.
I do think you love each other. It’s not that your mom doesn’t care about you. She is frantic about your grades. People who don’t care, don’t bother to fight. As for you: You do care what she thinks. And you are stressed out by the fighting.
The good news is that you both want the fighting to stop. I’m thinking if the two of you could solve the problem, you would have done so already. For that reason, I do think a few sessions with a family therapist would be very helpful. A family therapist can give you a safe place to talk about your worries and can help you learn new ways to talk to each other. Your school guidance counselor or your doctor can help you find a counselor who works well with mother-daughter issues.
I wish you well.