Transgender is the more accepted term. "Transvestite" depends, but it should only be used if you're in the community or have been given permission to by someone trans. I think that's what I've heard. But this is for the West, and I know Japan is a bit different? Like there's all that "okama" stuff that always gets translated as "transvestite" so I think the terminology and identity politics are different there.
I'm pretty sure the author used tranny purposely. A lot of times when someone is gay people think/try to demean them by treating it as if they actually want to be a girl instead. It's messed up.
a lot of people dont understand though, that the mangaka/translator is possibly using the word in a purposeful way to demean the character. a lot of rly young underaged readers(even older readers) just assume tht it's the proper way to refer to someone like that, and so i think it's important to at least point it out.
I'm kind of interested to the will be second couple (?) maybe, the tranny and the arrogant guy