As to the descent from Sarah, it’s a question of legitimacy. To the medieval European, descent from Sarah had primacy and descent from Hagar had less prestige. So to falsely claim descent from Sarah is to lay claim to a heritage that one does not have a right to.
Sure, but that’s very different from talking about a “slur,” which makes it sound like it was the medieval equivalent of the n-word.
This has me wondering why the Greeks would borrow an Arabic word meaning east instead of using their own, like ανατολικός. If it is because that is what the Arabic-speakers called themselves, then I have to wonder why the Arabs would refer to themselves as “sharq”, that is, “easterners”. East of who? Understand, I’m not questioning the veracity of the etymology, but only curious about how it came about. Perhaps it was as simple as some Greek guy asking, “What do you call yourselves?” and some Arab guy, thinking he is being asked where he is from, pointing east and answering, “Sharq”.
The Muslim world was divided into a western half, called maghrabi ‘western,’ and an eastern half, called sharqi. I’m guessing this is at the base of it, but of course I’d be interested in learning more about the history of the term.