No One Understood Transgender Tarzan
Saying that a person is transgender is like saying that a person is cisgender, it puts them in a general category but doesn't say anything particular about them. Is a cisgender person a man or a woman? Yes. Is a transgender person a crossdresser, a transsexual, or something in between them? Yes. We need to stop thinking that the term transgender says anything particular about anyone because it doesn't. If you want other people to know the particular truth about you then you will have to communicate it with some extra words. Use pantomime and sign language, if there are no words to express the real you. Just don't think that transgender does it alone. It's like saying that a person is a human being. It distinguishes us from nonhumans but not in a very meaningful way. It lumps all the disgraceful human beings with the good ones, and no one wants that to happen.
P.S. Kaitlyn, you are sweet to make allowances for my petty offenses. I appreciate it, even if I sometimes deserve a small rebuke. I think you gave me one earlier, but can't be sure. Your slyness made me chuckle. Thanks. I'm sympathetic to your point of view, but biased toward my own. No harm done, I hope.
Gender Curiosity Killed the Cat
I'm going to try to say this delicately. Cisgender people don't want to be lumped with transgender people. More to the point, cisgender people don't want to be lumped with crossdressers and transsexuals, the transpeople who most want to be associated with them. If you are a TS woman, try getting into the FAB section on this site, if you need proof. TS men are likewise blocked from joining the MAB section, I think.
I support transgender and cisgender differentiation, but also recognize that some trans populations are closer cohorts with cispeople than with some transpeople. For example, ciswomen and transwomen are closer cohorts than transwomen and crossdressers. However, transwomen will not be able to nurture this close association for their maximum benefit by denying their trans status. Cisgender people will not forfeit their special status without a fight that will never end.
I don't think anyone thinks of themselves in general terms most of the time, even though general terms apply to us. We think of ourselves in the most personal terms that are comfortable to us. I think it is wholly appropriate for transwomen to think of themselves simply as women since woman is a generic term that includes ciswomen and transwomen. If being a woman is more important to you than being a transwoman (although you are both) then there is nothing wrong with expressing this preference in the way you think about yourself. Transwomen who don't differentiate themselves from ciswomen at any level of meaning, however, are being dishonest and are not likely to find acceptance of this point of view among many of the cohorts they value most.
These topics should be thought about carefully but not constantly. Gender algebra is exhausting. Once you have solved your issues then stick to your final answer, and put the proof away in a drawer. Drag it out again if memory fails or someone important calls your answer into question. In other words, there is no need to justify yourself all of the time to everyone. Pick your battles wisely.