While reading Procopius's Wars (One side Greek, the other English) I came across the word Γραικουσ. Now I don't speak Greek but I know the letters and can sound them out well enough to recognize a few words.
I always had it drilled into my head in history and classics that the Greeks called themselves Hellenes, so why would an author writing in Greek use the word Γραικουσ ?
Here is an image of the page, the word Γραικουσ is used in paragraph 40
Oh boy, there's actually A LOT I can talk about here, but I'll keep your specific inquiry brief and leave the rest for any follow up questions.
In short and in the context of general Byzantine usage (especially later on in the tenth to its end), Γραικός is meant to act as a distinguish-er between themselves and the people to the West. They consider the Hellenes of the West to be completely separate from themselves. Even the Latins in the West didn't think the Byzantines were Greek. ------> Gounaridis, Paris. "'Grecs,''Hellenes,'et' Romains' dans l'etat de Nicee." Aphieroma ston N. Svorono 1 (1986): 248-57.
If you want to read more on what Greek identity is in Late Antiquity, around the period when Procopius is writing. Check out Bowersock's book on the matter. Bowersock, Glen Warren. Hellenism in late antiquity. Ann Arbor: University of Michigan Press, 1990.
EXTRA: Also, it seems like you might be shocked to hear that word used to describe Greeks. It's been done before and much earlier by Aristotle: Aristotle, Meteorologica A, 352b.1-3: οὗτος γὰρ [ὁ ποταμὸς Ἀχελῶος] πολλαχοῦ τὸ ῥεῦμα μεταβέβληκεν· ὤκουν γὰρ οἱ Σελλοὶ ἐνταῦθα καὶ οἱ καλούμενοι τότε μὲν Γραικοὶ νῦν δ’ Ἕλληνες