Would you say the former category are also men because they present as masculine?

Tomboys are still women, but if someone goes all the way and gets surgery, lifts weights, and gets a beard transplant, then it's easier (more pragmatic) to just call them a man. This doesn't mean that they are literally a man, but rather that they are in a sub-category of man, those who were born as women. More precisely they are trans-men, but the prefix can easily be dropped like many other words that are shortened when the appropriate context exists.

I would quite strongly disagree that there is such a thing as being "more of a woman" or "less of a man" based on conformity with gender stereotypes. This exposes the oppression of gender and my hope is that one day people won't be blackmailed into thinking they're less of a man because they're not an athletic lumberjack.

Yeah I don't think people should assign a moral value to being more of a man or woman. They should just be descriptors/archetypes that perhaps is rude to box people in. For the purposes of arguing whether a trans people should be call what they want to be called, I think letting men be feminine is already opening the door to this possibility. If a man can be feminine, why can't they go full feminine, including makeup, cosmetic surgeries, etc.? If we're ok with stretching those archetypes, then in theory it would be ok if some men are extremely feminine to the point that for practical purposes in society we call them a woman. The very important caveat here, is that everybody would know that they are really a trans-woman, and that you drop the prefix to be polite.

In other words, unless we actually oppress people into strict gender role boxes (shaming gay men, etc), we're allowing for trans people to be part of society regardless of whether it's a mental illness or not. But stretching the archetypes doesn't mean that we deny reality when the details are necessary.