People can call themselves whatever they want, but if we don't have consistent definitions of terms to use in our communication, then it doesn't help or may hurt all of us in the long run. The vast majority of cisgender people, even those who are sympathetic to us, have extremely simplistic understandings of our issues as it is. Introducing confusion is a barrier to acceptance, not an enabler of it.
First of all, "trans" is not a word. It's prefix. And by itself, it doesn't mean anything specifically in regards to gender identity issues. "Trans*" is also not a word; it looks more like a logo or brand name. I don't understand the people who insist that "transgender" refers to a concept that's separate from "transsexual," "androgyne," "genderqueer," "third gender," "cross-dresser," "->-bleeped-<-," etc. "Transgender" works as an umbrella term because it doesn't have a strong definition on its own.
"Gender-variant" and "transgender" do not mean the same thing. "Gender-variant" might have very little to do with being transgender. We can all agree that gender largely consists of a package of conventions. But let's look at the word "variant" or "variance." It doesn't mean "being something that's completely different;" it means "having differences without going outside of boundaries." A gender-variant person is someone who has traits or behaviors that are outside of the conventions of that person's gender-assigned-at-birth but does not otherwise identify as being a different gender. She or he believes that gender should be defined more broadly and be more inclusive of others who are different, like herself or himself. An example would be a tomboy. Most tomboys don't think of themselves as androgynous or FTMs; they think that men shouldn't have a monopoly on certain behaviors, interests, fashions, etc., and that being butch doesn't make them any less female.
Being transgender means breaking gender boundaries. If you're a cross-dresser or ->-bleeped-<-, then you accept that what you do crosses gender boundaries, but you don't think that "femininity" or "masculinity" or "women" or "men" should be defined more broadly or differently. It's a largely compartmentalized part of your life that doesn't give you crises about your gender identity (other than any guilt caused by disapproval from other people or society's ideas about gender expression). If you're an androgyne or genderqueer or third-gender, then you reject the boundaries implied by the gender binary and instead identify as both or neither (although that's not exactly accurate, since "both" and "neither" imply a certain acceptance of the binary). If you're a transsexual, then your body is the boundary, or it's a boundary on top of other gender boundaries, that must be crossed. This doesn't mean that you have to do a full transition with SRS, but it does mean that your body is a fundamental part of your dysphoria. Some people feel that none of these labels apply to them, but "transgender" still seems right, and that's fine. But that doesn't mean that those labels don't fall under the "transgender" umbrella.
None of those alternative gender identities is mutually exclusive from the others. I personally identify as transgender (using it as an umbrella term), transsexual, and also androgyne with a hint of bigender. For the sake of simplicity, I prefer the phrase "gender fluid." I'm also somewhere between bisexual and pansexual, but that's another story...
