In America, who is white? The General Social Survey has asked people over the years where their ancestors came from, and they routinely ask respondents what race do they consider themselves to be--white, black, or other. You might expect that no Japanese, East Indian, or American Indian would answer anything but "other", but you'd be wrong. Look at how individuals from different groups see themselves racially:
Percent who see themselves as a particular race
Clearly, many people are describing the color of their skin; that is what they understand "race" to mean. But I suspect that another factor is how you see yourself with respect to Americans descended from Europeans. If you identify with them, you are more likely to say you are white.
If people were merely describing their skin pigmentation, I would have expected more Greeks, Arabs, American Indians, and Mexicans to report "other." I'm sure many "American Indians" are whites who are so proud of their small amount of Indian ancestry, they claim it as their ethnicity.
Notice how almost no one whose ancestors were from Africa thinks of himself as white, and how quite of few blacks think of themselves as "Americans only", with no connection to Africa whatsoever.
Most important for this blog, most Mexican Americans see themselves as white, which is a positive sign that they identify with white Americans and want to integrate with them.
Update: I added American Jews to the bottom of the list. There is no indication that some in this group want to distance themselves from whites by saying their race is "other."