Am I a native American or not?

I was born in Oklahoma., raised here, born of parents who were both born in Oklahoma and raised here. They were born of parents who were born in Missouri and Oklahoma and raised in Oklahoma, clear back to the late 1800s.

I am a native Oklahoman. And I am a native American. But I am not born of an Native American (Indian) tribe; well, sort of. My paternal grandmother claimed to be a quarter Cherokee on her paternal grandmother's side, although that family was not listed on the Dawes Indian rolls. So we have no proof of it.

Grandmother's two sisters married full-blood Cherokees and all of their children are on the rolls. So my distant cousins are native American.

In a way, I am a native American. A native is one who is born and raised in the country. And an American is one who is born and raised in America, so how does that relate to a "native American" being defined as a person of an Indian tribal heritage.

I understand the desire to differentiate those of that origin, but I also wish to be known as a native American myself.