It's not so much the gender role or social norms that matter--not all effeminate men feel that they should be women, and vice versa.

By saying "incorrect genitals," I was simply using a shorthand for "incorrect biological gender," or a biological gender which a person does not identify with.

That sentence could be rephrased as "Transsexuals were born with the biological gender, compared to their psychological gender."

(N.B. I'm not sure if "psychological gender" is the commonly used phrase, but I think it's adequate to communicate my meaning.)