My ethnicity is widely varied due to the melting pot this country is based upon. There's some American Indian in there a couple generations back and certainly not enough to be considered for membership in the tribe my mother claims was in her father's lineage. I wouldn't try to claim it unless I was close to "full-blooded" but I also think it has to do with your interaction with that group. I engage in activities and rituals with a couple reservations in my area and am accepted as family in that when I go to any of those places I am greeted with "welcome home". which I find humbling and greatly respect.
So in tracing this lineage, it normally seems to go back as far as one or more blood relatives who were known to be "full-blooded" and go from there.
It sounds like a curiosity of yours and I think it's a worthy point to investigate, since you are wondering about... check it out.
Looking at the info you posted below, it seems there is a question as to the actual status of Hawaiians but I think any aboriginal group whose ancestral home has been annexed into the US is worthy of the title, and many of my Indian friends feel the same about that.
My two cents.
FWIW..
In the end, we are all members of the human race with endless variety... like language, there are many varieties in most languages and English could be seen as a good example here. In anthropology, there is only one race.. the human race.
Thanks for sharing and asking an interesting question that prompted me to think about this... I have friends from all over the planet and I find it fascinating how different ethnic groups interpret this question.
Good luck in your search for an answer.
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