As a European, I am confused about the use of names for ethnicities in the US. The background of my question is academic (I usually work with quantitative data on ethnicity on other continents, mostly about self-determination organizations and parties) so it does not matter how poeple self-identify. For example, I gathered that only certain people are considered to be native Americans. But are terms like "Asian American" colloquial, or does it refer to the Asian continent? What about people from Hawaii or Puerto Rico?
For academic work, the terms do have definitions - managed by the Census Bureau. They do have colloquial uses, of course, but if you're looking at quantitative data then there are definitions which are sample-validated.
For Asian-American, since you specifically cite it, there's a number of geopolitical issues in play.
The Census definitions, however, may be found here, and generally will be supplied as part of the metadata for any official dataset you can find. Datasets that do not come from the government, and do not provide alternatives are extremely likely to be using these definitions (or whatever definitions from the Census Bureau are contemporary with the data collection effort - these are subject to change) in order to keep their datasets comparable to census-derived data.
Deciding on names of ethnic groups can be a political issue. The use of language affects the way we think, and vice versa, and so people try to affect the way we think by trying to influence how we talk.
The United States is largely a society of immigrants. More than a century ago, it became common to use the place of origin with a hyphen and American to describe many immigrant groups, thereby acknowledging their American identity while drawing attention to their non-WASP-ness. The term Asian-American describes immigrants from Asia and their descendants. Note that African-American or Afro-American only slowly replaced racist slurs in many groups, and that instead of European-American there are more detailed terms like Irish-American or Italian-American.