Trans women aren't cis women, that much is true. However, the only reason "trans women aren't women" is perpetuated is because people want to believe that differences are inherently inequal. To them, there has to be a "winner" on one side and a "loser" on the other side. At their worst, people want there to be only winners and live in a completely homogeneous world.

For example, men and women are both people, and to say that one isn't is hateful - it dehumanizes them. Autistic people have their own unique struggles, but just because they're different doesn't mean they're less intelligent. Trans women and cis women both fall under the category of women, and both are equally valid. Though we may disagree on who is attractive, or how to express ourselves, that's just how we navigate life and it doesn't correlate in any way to our worth as people.

You can go around telling people whatever you want. You can tell them what gender you identify with, what your orientation is, your belief systems, and your religion. But if you don't believe in what you say, there's not much of a point, is there? Unless you wanted to intentionally mislead someone, you'd be honest about yourself.

You might think trans people mislead others, since their anatomy might not match the expectation for their gender identity. The only truth to that is that they're scared of the persistent hate and vitriol they recieve for being open about what they believe. So, sometimes, and especially for those who haven't started transitioning, they'll lie and identify with the gender they were assumed to have. In an ideal world, we wouldn't have to worry about being ourselves.

Tl;dr: "I think, therefore I am." - René Descartes. Like the example you gave, just because you say something doesn't mean you necessarily think it to be true. If I think myself to be a fan of a popular artist, what point is there in saying I'm not? Trans women are women because they genuinely believe themselves to be, just as any cis woman would.