"American" is the term in the US because it's not tied to ethnicity. "Chinese" is tied to ethnicity.

English speakers may think of "Chinese" as an ethnicity, but it's different in Chinese. The default term is 中国人， which literally means "person from the middle country", which is quite obviously about nationality. If you want to talk about ethnicity, you'd use 汉人 for Han Chinese, 蒙古人 for Mongols and so on for any of the hundreds of Chinese ethnicities.

“American” suffices over here. You need additional qualifiers to distinguish native-born from naturalized.