From wikipedia:
"The name Oklahoma comes from the Choctaw phrase okla humma, literally meaning red people. Choctaw Chief Allen Wright suggested the name in 1866 during treaty negotiations with the federal government regarding the use of Indian Territory, in which he envisioned an all-Indian state controlled by the United States Superintendent of Indian Affairs." [my bold]
“Choctaw Chief Allen Wright”
Looks like Indians began taking Anglo-Saxon names long before the present age.
Much easier to say than Walking-Eagle-Who-Dances-with-Wolves or whatever.