transwoman	born as a girl, identifies as a boy
transman	born as a boy, identifies as a girl

I'm pretty sure you wrote those two reversed of what they are commonly defined as.

Regarding the hysterectomy, I assume like with any surgery it comes with a lot of added risk and of course additional cost. But would it not also reduce risk, of cancers for example? Is there anything else in the trade-off that I am missing?

So would it be fair to summarize your definitions of transsexual and transgender as the following? A transsexual person is someone who is suffering from or who used to suffer (but no longer does because they received the appropriate medical attention necessary) from gender dysphoria. A transgender person is someone who has strong preferences for certain behaviors and outward expression that differ significantly from those that would be stereotypically inferred by most of society based purely on the birth genitalia of the person in question.

I tried to avoid using the phrases "man," "woman," and "gender identity" in the above definitions, because honestly those words are so vague and unclear to me within this context that they may as well be meaningless (unless you can properly define them).