You are in a very tough situation. If your mom is like most, she just wants the very best for her family, and oftentimes when we are making decisions our personal experiences will trump evidence which contradicts what we have lived through ourselves. As you continue your efforts to go to boarding school, just remember to remain thankful that you have a mom who loves you so much!
Echoing the thoughts shared by @atria above, it sounds like your dad could be your strongest ally. What does he say about convincing your mom? If you are really serious about this, I think you need to work with him to help you figure out if there are ways that both of you can help your mom accept and support your goal of attending boarding school. I also have the impression that boarding school cultures have changed a lot since the days when your mom attended high school. Helping her see these changes may help too.
Something to consider is that deciding to go to boarding school is very different from deciding to apply to boarding school. When our family started our boarding school journey, AppleKid was only pitching the idea of applying, and continued to investigate local options as well. Maybe your mom will support you in going through the investigation and application process (it sounds like you have already started your applications?). As you will gather from what you read on this forum, being offered a spot at a boarding school depends on figuring out and applying to the school(s) where you would be considered a good fit. Truth be told, AppleKid, DH and I were very far from certain of getting offers of admission from the schools to which AppleKid applied. So we really didn’t have to face the decision of whether or not to let AppleKid go to boarding school until M10 (when decisions are sent out by most schools). Assuming that you want to apply to highly selective schools, that’s pretty much the case for everyone no matter how smart, charming and accomplished they may be. 