Note: I'm not trying to be rude, but I find your post frustrating. If you're going to make what I perceive to be somewhat ignorant comments and really, what I construe as prejudice remarks, please actually read what a post says. We are discussing whether or not the OP should put down that she held a leadership position in a LGBQT organization, which is very much the same as holding a leadership in any other organization (which is why I mentioned sports, music talent, etc.). While you don't absolutely need to mention it specifically, I think it's pretty normal that if you put down you were a leader of a LGBQT organization you might mention a motivation for being a part of it....like for instance, you being transgendered, gay, lesbian, etc. It's just as if you put down that were president of Asian Student Association, that the committee would instantly think you joined the organization because you yourself were asian and it was a part of your identity. Also, your assertion that your own identity has no bearing on why you want to be a physician is ridiculous.
Suppose an applicant grew up in a community of African immigrants (herself being one) and saw problems with her friends and family getting healthcare and/or communication with physicians. That provides her a motivation to become a part of the health system because she wants to serve her own cultural group (which she is a part of). It the same for people of alternative sexualities and gender identities, all who live in a unique social sphere structure, have unique illness attitudes and medical seeking behaviors, and sometimes desire professionals who share these ideals in what we pretty much all agree is a traditionally conservative profession. Your med school app tells adcoms not only why you'd be a good physician, but also why you personally WANT to be a physician. So, yes, putting down that you're gay in the context that you've had substantial leadership experience in this field and possibly, that it motivates you to become a physician, is highly relevant to a medical school application.