I'm an ethnic minority (Chinese) and an international student (Singaporean), so I'm not sure how well I can relate. I do think that your experience probably depends a lot on your field and program, and I'm happy to hear that it's gone well for you so far.
I can sort of understand what it feels like to be representing a specific group (in my case, country), but obviously race is a lot more salient in the U.S. That said, on purely rational grounds, nobody has the right to burden you with the responsibility of setting an example. It's your life - you should succeed and fail on your own terms.