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.