My impression is that most Americans believe that to be rich you have to have a butler, like on Fresh Prince of Bel-Air.
Worldlier Americans believe that to be rich, you have to have a private jet. For example, I suspect that Bill Clinton and Al Gore don't consider themselves rich, precisely because they are always having to bum rides off billionaires and dictators with private jets. (Or have they cashed in enough by now to buy their own?)
Ergo, almost no Americans see themselves as rich.