Not only does she believe it, it seems to be fact.
There is some evidence that it could be true. I don't think it is close to being established as a fact.
In any event, this is all dumb. I hate to say this, because I've been on the Warren train for years, but she handled this badly. All she should have said, which is what she had said in the past, is that she was lead to believe as a child due to family stories that she had Native American ancestry. All she had to say is what was established by the Boston Globe:
The Globe closely reviewed the records, verified them where possible, and conducted more than 100 interviews with her colleagues and every person who had a role in hiring decisions about Warren who could be reached. In sum, it is clear that Warren was viewed as a white woman by the hiring committees at every institution that employed her.
The evidence is clear. She never listed herself as anything but white in any of her admissions documents, and only later changed her ethnicity designation, around the time her aunts, the matriarchs in her family, started making their supposed Cherokee background more of a prominent thing.
She never once claimed to be a member of the Cherokee Nation. This was always an issue within her family. And it seems to have stemmed from a sincere belief regarding how Warren's parent's had to elope because, apparently, at least her father's family thought that the claims of Cherokee ancestry were true enough to disapprove of the wedding.
I think she fell into Trump's trap here, unfortunately.