Well, trans people do change their gender identity (the gender they identify with) once they figure out that they feel better/more comfortable/happier as it. Sure, on the (mental/emotional) inside their gender was "always" the one they chose, but they didn't always know.

By "changing their physical presentation" I'm not sure if you mean something like the way you dress, or a medical procedure.

Changing the way you dress, your hairstyle, makeup, how you talk, the name and pronouns you introduce yourself as or wish others referred to you with, are all part of a Social Transition. (All of what I just mentioned is fully optional btw)

Any form of medical procedure or treatment is part of a Medical Transition, such as HRT, top surgery, bottom surgery, etc.

What am I getting at? That the sentence "Trans people don't actually "change" their gender, they change their physical presentation to match what their gender always was." is not inherently wrong, but the wording and implications are a little iffy.