Why care about being labelled as "integrated" or not?

I don't know there is such a public, agreed upon standard to define who is "integrated" or not, why care about the labelling from someone or else, even he or she is famous?

And I don't even agree that the native born American (or Europeans) are good candidates to judge whether a Chinese (or Indian or whoever) is "integrated" into American (or western) culture, just like I don't think Chinese are good at judging who is "Chinese Expert".