For what it's worth, most people I know are fine with "American Indian" as opposed to just "Indian," especially in contexts where it could be confusing. In written communication or in spoken conversations where it's really clear what peoples you're referring to, simply using Native is also pretty common, at least among people I know (preferred terms can vary a lot by region and even age). It's easier in written conversations since the capital N sets it apart as a proper noun.

Using the specific tribal affiliation is definitely the best choice if possible, though. It's kind of comparable to Europe. You can say that someone from Sweden and someone from Greece are both European, and it would be true, and it can sometimes be useful to group them all together. But if you said that Europeans live in an area with harsh winters and love their saunas, that would also be kind of true, but it also doesn't really include the Greeks (or Spanish, or French, or Italian, or...). Except everyone knows that there's a difference between Greek and Swedish cultures, but decades of stereotyping have made it so that a lot of people don't really realize that there are equally large differences between various Native cultures.