Yes, they see your name.
Compared to Ivies and top research universities, Asian applicants at LACs are a much smaller portion of the applicant pool. Presumably this is due to the name recognition of the Ivy & research universities in home countries, and parental pressure as well to go to a college that will be "recognized’ by the folks back home.
@ucbalumnus, Asian enrollment is not the same as the applicant pool, of course. They may be underrepresented in the applicant pool, but accepted at a higher rate than non-Asian students (partly due to high stats). In general an Asian student applying to Swat is going to have a better chance than at most of the Ivies. The top colleges do limit their offers of acceptance to over-represented groups.
So OP, my advice is don’t check the box with your ethnicity at Ivies and top research universities. Go ahead and check it for LACs. Probably doesn’t matter for universities that aren’t in the top tier.