A transsexual person is someone who has undergone bodily modifications to more closely resemble the opposite sex.
A trans-identified person is someone who conceptualises their sexed identity under the rubric of trans ideology. It includes transsexual people as well as those who have undergone no or minimal bodily modifications but still wish to be regarded as the opposite sex, and people who claim to be neither male nor female ('non-binaries').
These terms exist because trans-identified people exist and we need to have some words to describe them. You seem to be very hung up on the idea that the cultural phenomenon of transgenderism is all about weird people being weird, whereas I think it is inextricable from wider cultural beliefs about sex and gender.
I agree that we need to be very careful with how we use language. I use these terms because IMO they are the most precise and most likely to bring clarity to the debate, although I grant that 'transsexual' is a misnomer. Nevertheless since it refers to an objectively definable category of persons (unlike 'transgender'), I'll use it.
'Gender non-conforming man' doesn't work as a term for a man who identifies as trans, because any man can be gender non-conforming. Also, most trans-identified men are NOT gender non-conforming. A truly gender non-conforming man rejects male dominance and entitlement to women's labour and deference, which most TIMs certainly do not. It is precisely because I am careful about language that I never refer to TIMs as 'gender non-conforming'. And I have zero problem with referring to TIMs as men (which was the original topic of this thread). However, sometimes it is necessary to distinguish them from non-trans-identified men.