"Native" names and "western" names?
Original name and alias/adopted name are generally used.
I would use Chinese name and English name. Note, not his Taiwanese name nor his American name. I would use the language, not the country. e.g.:
My grandfather, Shachna came to America. "Shachna" was his Yiddish name, but "Charles" was his English name.
(Note: Jews also have a separate calendar and often people will say "Hebrew date" vs. "English date" which is wrongly derived from this. It really should be "Hebrew date" vs. "Gregorian date".