I think that white/Caucasian in its current sense (and Caucasian particularly is a really fraught term), is almost always going to mean 'of European descent' -- and even if 'white' specifically doesn't denote that, that is the history that's taught about Australia, North America, other places with significant white populations. That doesn't go into much detail, but neither does Asian or African. I'd say they're all roughly equivalent. I agree that white people are by no means a homogeneous group, the chequered history of Europe alone being proof enough, but I think all ethnic and racial identifications suffer from the same problems -- when you've been somewhere long enough, you don't know that much about where you came from. I don't think there's much danger of anyone not being aware that the white people of the US were originally European migrants.