According to the Oxford English Dictionary on the etymology of the term Anglo-Saxon came later:

“The term was first used in the early 1600s, with the earliest known use recorded in 1602.”

“Angle

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noun (2)

An·​gle ˈaŋ-gəl

: a member of a Germanic people that invaded England along with the Saxons and Jutes in the fifth century a.d. and merged with them to form the Anglo-Saxon peoples”

This is as far back as I can find with

Miriam Webster. And then this from some random site:

“A famous anecdote, though likely apocryphal, recounts Pope Gregory the Great mistaking the blond, blue-eyed Angles for angels, saying "Non Angli, sed angeli" ("Not Angles, but angels"), which highlights the phonetic similarity between "Angles" and "angels" but does not imply a shared etymology.”