The Executive Editor of Indian Country Today answers the question.
While it is true that the term “Indian” does not accurately describe the indigenous peoples of the Western Hemisphere, its usage, particularly when incorporated into the term “American Indian” has been largely (although not universally) accepted by most tribal officials. Also, as a matter of style usage the term “American Indian” seems to be gaining favor over the term “Native American,” since it carries with it a more specific identification. For example, anyone born in the United States could technically call themselves “native Americans,” but cannot identify themselves as “American Indians.” Our newspaper prefers the term “American Indian,” and we will often use the single word “Indian” for subsequent references within a story. Although language evolves through time and usage I would suggest that the term “American Indian” will likely become more commonplace nationally and internationally. One factor is because national publications such as ours, Indian Country Today, have made it our style choice. Other factors include its use in nationally and internationally renowned organizations such as the Smithsonian Institution’s National Museum of the American Indian and the National Congress of American Indians, etc.
When I lived and worked on the Rez in Central Arizona, interacting with members of various tribes, they always referred to themselves as Indians. I don’t think I ever once heard any of them refer to themselves, or others, as Native Americans.