It's ultimately your responsibility to take control of your own life and your own presentation and to correct people when they misread you. But, forgive me, you appear to be sending mixed signals. It's probably unavoidable because you aren't full time yet, but are you absolutely sure that your friend fully understood your situation?
First you say that you go by your birth name at school. All well and good. But this friend seems to know you outside of school as well. If that's the case, she's living with one foot in one world and one foot in the other world, so to speak, just as you are. That can be confusing, as I'm sure you're aware.
Then you say you don't mind going by both names for now...does that mean that you go by one name in certain venues and the other name in other venues, or does it mean that there is some crossover? If there is any crossover at all, can you blame your friend for getting caught up in it?
Then you say that at long as your friends respect your persona, you don't care what they call you...but do you have a problem when the friend calls you "Kyle" and "he" at school, where other people apparently do not?
Then you say that you aren't out at school unless people ask, and then you tell them...this friend obviously knows that you are out, and she treats you as Kyle. Did you ever sit down with her and explain how she is to address you at school, where you are not generally out except sometimes to some people?
I'm not trying to flame you here, not at all, but you are in that awkward not-fully-out stage--which is a challenge for anyone to negotiate. And I guess your main concern was whether your friend should have been so adamant. I really do think that this kind of confusion was created by two worlds in collision. And I think that your friend was overzealous; but on the other hand, she seems to be in the middle of those two colliding worlds; did you ever speak to her about how to address you at school?
I'm glad you talked to her and received an apology. But what if she hadn't gotten overly enthusiastic and had just called you "Kyle" and "he" a couple of times? Would that have been a problem when the other people were calling you "birth name" and "she"?