The attempt at a “gotcha” is annoying. I can only speak from my lived experience as a white person who grew up on a reservation. I married a native, and almost all my closest friends were (and are) natives. Everyone I knew called themselves “native” or “Native American”. “American Indian” wouldn’t have been considered offensive (unlike “redskin”), but it really wasn’t the common parlance. The only place one would routinely see the phrase “American Indian” is on government paperwork, eg from the BIA.
This is one of those things you can find out for yourself: go to a pow-wow (non-native visitors are typically very welcome) and ask the people there how they usually refer to themselves.