“What is a "transgender"?”

The meaning varies depending on who you ask.

What is a "trans man"or a "trans woman"?

Simplest explanation- someone who undergoes the process of altering their appearance to resemble the opposite sex

“A "transgender" (person) is someone whose personal identity does not correspond with their sex, and who wishes to change their "gender" from "man" to "woman" or from "woman" to "man" through hormones or surgery.”

Sure, if we accept that your sex means you have to or aren’t allowed to identify a specific way.

“This is confusing because, there is an assumption that "gender identity" exists,”

An assumption made by people who want it to exist, despite no clear explanation for what it means, what if feels like, or why it means different things to different trans people.

Answer, please:

a transwoman knows he’s a woman/has a female gender identity because he felt “out of place” in his body, is drawn to “feminine” things, and didn’t connect with what other boys/men tended to connect with?

another male who likes the same things as the first, but is “comfortable” with his body and his sex,

And also a female who feels “comfortable” with her body and sex, but just wasn’t interested in traditionally “feminine” things

Do they all have the same gender identity? Why or why not?

“and that "man" and "woman" are "genders"”

the genders are male and female, they correlate to sex. Intentionally. Because that’s the whole point of gender.

“or defined based on "genders" instead of sex,”

Gender is based on sex, you can’t really define something based on gender without it ultimately being based on sex. There is nothing that a trans person can do post transition that they couldn’t before, except experience some of the side effects of gender the opposite sex experiences- if - they successfully “pass”.

“which can be changed through hormones and surgery.”

Humans can’t change sex, that’s just fact, so we shouldn’t define sex with falsehoods or by the ability to change the appearance of your sex, but not the fact of it.

“Both contradict GC's view that "man" and "woman" are sexes instead of "genders", and can not be changed.”

Men and women aren’t sexes. They’re the terms used to differentiate between adult humans of either sex.

“Would it be correct to conclude there is no such a thing as "transgender", "trans man", "trans woman", and "transitioning" then?”

I mean... yeah? Kinda? That doesn’t mean that there won’t be people who alter their appearance to appear to be the opposite sex. It just means that gender is not some internal sense of self or something you can opt into, it is just society’s interpretation of sex. You can’t turn into or out of it, you can only successfully slip undetected into the structure that already exists... or have no need to. You’re ability to slip by undetected relies on sex meaning what it actually means, and gender being directly tied to sex.

It doesn’t mean that the terms “transwoman” “transman” or “transition” can’t be used by people who wish to, or when necessary for discussion, it just means that when it comes down strictly to fact, transwomen are men and transmen are women. And these terms used to avoid stating that fact are used kind of as a societal compromise.