You mention that trans women should be seen as women. As a trans woman myself, I appreciate that sentiment. However, that raises the question of what you, as a woman who is not transgender would like to be called if you do not like being called cis. Just a woman? Well, I’m a woman, too, as you pointed out and calling yourself a woman as opposed to me would imply I’m not. Does that make sense? A non-trans woman? Well, cisgender literally means “not transgender” (cis is the opposite of trans in Latin, as has already been pointed out). If you absolutely do not like being called cisgender, then I have no problem calling you non-transgender, but it seems a bit clunky.
This is a serious question, and not a gotcha — what substitute would you accept for the word cisgender? If it does not imply that trans women are not women, I’m sure most trans women would be on board. As an aside, someone calling herself just a woman, or a “normal” woman would not achieve the objective of differentiation without dehumanizing or othering trans people. I’m no less normal than many other kinds of women, and certainly any lack of normality springs from just about every source other than my being trans.
I hope the points I have made make sense to you and that they are worth some consideration.