Not everyone with dark skin is "African American"; not all of them are Americans (of course!) and not all of them are Africans. More significantly, though, not everyone who is African American is necessarily black. You can't say that Nelson Mandela is African American: he isn't American. OK... so he's African then? And what about P. W. Botha? Isn't he African too? He was born there, right? I have friends who have moved to America from Africa who are not black: it does happen.
So the problem is that "African American" does not equal "of dark skin." Which perhaps is the point: to put the focus on culture rather than color. And as such it makes a certain amount of sense: there is something to referring to African American culture as a thing unto itself--as distinct from other American cultures, and as distinct from African cultures or cultures developed by black people in other parts of the world. The problem is that because of overzealousness in political correctness, and various applications of "USA=World" thinking, it has come to refer to the color and not the culture, leading to such sillinesses as "British African-American" above. So it no longer means what its components mean, and has become nothing more than a code for what they were trying to avoid in the first place. Bleah.
I guess the problem is that people insist on trying to classify folks by color. The term "African American" attempts (weakly) to stop people from doing that, and make them focus on culture instead, but people blithely co-opted the term to keep to their old habits and in many ways wound up in the same boat all over again.
On the flip side, sometimes things can get pretty silly with avoiding "black." There's nothing inherently wrong with color, and pretending you can't see what's in front of your face is silly. A friend of my father's tells of the time he asked someone if the lawyers for a case he was dealing with had been by yet. "Yes, one of them." "Which one?" "Well, the tall one, with hair like..." "Look, was it the black one or the white one??" It's a fair question, not a racist one, just asking for an easily-seen feature, but the person was so conditioned to be "color-blind" that he couldn't give the obvious helpful answer.
Lastly, things change with time, of course. I had a friend (rest her soul); she liked to think of herself as my third grandmother. She was an old shut-in I used to call and talk to, and she was of the "old school": she didn't like the fact that she couldn't find the right check-box for herself on forms: she didn't like the term "African American," she didn't like "black"; she wanted to call herself an "American Negro," a term she felt good with (though most people these days would not).