I have always found the term “African-American” (or Asian-American, or Mexican-American, and so on) to be kind of weird – almost a forced “politically correct” (and I don’t use this term lightly) thing people use to try really, really hard not to offend almost to the point of absurdity. I can’t tell you how many times I’ve seen Americans on TV in foreign places saying a black person is “African-American”. The reason I find the XXX-American concept strange is that it’s trying not to offend while making various assumptions about the person that may not be true – they may not even be American, and they may not identify with any ancestry from X country or region.
If someone called me “European-American” I wouldn’t necessarily be offended, but I would find it very curious for sure. I certainly wouldn’t feel any special kinship with someone native to any of the regions of Eastern Europe that my ancestors came from, nor would I identify as being “European”. I’m just an American white dude (feel free to call me a “stupid American”, I promise I won’t mind).
You are right though, in America, “black” has a very specific connotation – and that’s of someone that has features commonly associated with the people of sub-Saharan Africa more so than skin tone. Even someone who’s light skinned to the point of appearing Caucasian would probably be considered “black” by many. Dark skinned South-Asians for example would just be called “Indian” (because who the fuck cares what country they are from, right?).
As for sensitivity. I’ve had many black friends throughout my life (and no, not in the “I’m a white guy who has black friends so it’s OK for me to use the ‘n word’” sense, I mean most of my best friends throughout my life were black). Discussions about things like race and racism came up all the time because these were important and inescapable parts of our identity. Using skin tone as a shorthand designator was never considered any more offensive than saying “white”, “blond”, “short”, “tall”, or other obvious identifying characteristic about a person. (That said, if someone asked me to address them a certain way, I’d certainly respect this.)
Anyway I’m not sure if I actually made a point here or was just rambling on to the point of incoherency. As many have pointed out race, racism, and racialism aren’t things that can be solved just by carefully curating how you address someone’s ancestry.