I go by Moshe all over, credit cards, bank accounts, online accounts, etc. even my passport is like that, but my license says Moses, as does my SS card. It's a discrepancy that's never been taken care of because it never caused too many problems. I've flown on multiple airlines like this, never had a problem till now, I guess they understand it's the same name, just one is the biblical pronunciation, the other the phonetic Hebrew pronunciation. (Or maybe today with using my phone as the boarding pass by just scanning the barcode they usually don't look at the names so closely? idk)
How could it be a third party error if that's the way my name is on AA.com?