I like "American Indian" better. That gets it across. As for the word "Indian," the same is true for "Latino" and "Hispanic" as currently used in the USA. All three labels reflect a whole 'nother continent and a whole 'nother genetic pool, and it doesn't matter a damned bit. As for the word "Native" -- if you were born on American soil, you're a native American.
"American Indian" says it best.
Well I was born on American soil and I’ve been here my whole life, and I don’t have any allegiance to any foreign country. I guess I’m a Native American!
I once heard that Native Americans or American Indians actually prefer to be referred to by the name of their tribe, i.e., Sioux, Lakota, etc.