Speaking only about my part of the country, I can say that we really seldom even use the term "Asian", when referring to a specific person or group. It may well be different on our West Coast, which has the greatest influx.

As I hinted at in my other post, we here in the U.S. kind of need to take special care when using the word "Indian",
because of the long-term usage as applied to the ones who
greeted Christoher C. and Raleigh. People I know usually say "a person from India", or similar. If I heard someone described as Asian, it would not occur to me to think they were from India.
Overall, I'd say we tend to use the country of origin:
Chinese, Korean, Japanese. Interestingly, here the area containing Vietnam, Thailand, etc., is popularly known as
Southeast Asia.