How many of the 4 "components" of the picture do you agree with?
Did you use to believe "one side" until recently?
If so, what made you change your beliefs?
Do you believe that generally, your gender is part of your anatomy or your personality?
Gender = the left side (more or less)
Sex = the right side
Except for the third one. Neither sex nor gender are the same thing as sexual orientation. Essentially, statistically most biological males (whether this mean men, transsexual women, or gender-variant "males") are attracted to feminine biology and vice-versa; this is where the association (between sex and sexual orientation) ends.
(Slightly related question, what do people who undergo sex change operations mean when they feel like they're born in the wrong body? I have no idea what it would mean to feel like a gender or sex. I just feel like me, independent of sex. If I were born female I think I would end up being the same person. All I ever see on the media is someone basically going, "When I was growing up as a young girl, I didn't like pink dresses. So, now I want a beard so I can be the man I was intended to be". But dresses are a social construct and shouldn't define who you are. So, yeah I don't get it. Fill me in here.)
Some people (transsexual people and some other gender-variant people) have a gender identity that differs from their biological sex. This causes a feeling of "disagreement" often resulting in the desire for one's body to reflect their gender identity moreso than their biological sex. At the same time, though, sometimes people let gender roles into it too easily and base most of their thoughts or actions off of the stereotypical roles of the gender they're supposed to be. That's an unrelated matter, however, to simply wanting to grow a beard or look like a man.
Sorry if this sounded a bit eggheaded or technical, but basically (in the example you mentioned) the person has a male gender identity--something that goes beyond simply not liking pink dresses, and he wishes to appear male because it more fits the way he feels his body should be. This . . . is the best I can possibly word it. xD