Although tiresome to argue dictionary pedantry, I think the crux of it is that the implication is that the the US doesn't have any defined role in the rule of a

Although tiresome to argue dictionary pedantry, I think the crux of it is that the implication is that the the US doesn't have any defined role in the rule of a lot of the countries (as in, they don't claim ownership, or even influence), but that it's exerted either overtly or covertly by other means. The point in this context is that the US's web of influence extends outside of the areas in which they have a direct structure of power in place (and isn't consistent within those that it does) - those points, in this sense, imply an "Empire" (quotes, as nebulous), rather than an Empire (the defined thing).