"hermaphrodite" is terminology that is being retired in favor of intersex, as a slur that was also inaccurate/misleading.
"transsexual" is generally considered old and out of fashion these days, since it focuses on 'sex' which is, in fact, not connected. A bunch of old school people still use the term, but the younger crowd not so much.

The transition description there isn't out of place for a 'third gender' in some other culture, but it does not map to what a transgender person experiences and is kind of.. Odd really. I mean, what is the magic actually linked to exactly? Because transgender people are born transgender. If magic is biologically sex linked, it's just one more thing to be uncomfortable about not matching.

If people who change over are given special third gender status, I would expect to see a lot of "O Sacred one!" "Please stop, I'm just some girl." There has been a push in other places to highlight "We aren't <third gender>, we're just women/men!" And a small push for "Please just mark me as 'none of the above' without arguing". The third genders are often projected onto people based on formalizing whatever odd niche GNC people were able to carve out of necessity.