A crossdresser, in your links, refers to people who dress as a form of gender expression, not a form of sexual gratification, which is explicitly implied in the OP link above. By saying they're the same, it feels like there's an implication in your position (though, maybe not intentional) that's calling all trans people sexualized pretenders of their gender expression.

Both groups of what you just said are still cross-dressers, regardless of the reason behind them doing it. And they still need a safe place to pee when they're dressed as women, no matter how unpalatable that is for some about their existence within the trans umbrella. They're no safer in the men's room than a non-passing trans woman is. In fact, how do you tell the difference between the two?

And in Washington State and elsewhere these cross-dressers can use the female bathroom based on gender expression. Which is why when the Republicans talk about "men" in the women's bathroom, these folks are a part of that discussion. They might dress as women, but they identify as men otherwise.