Indeed. A Moor may not have been black historically, but Shakespeare obviously referred to Othello as black - "Her name, that was as fresh As Dian's visage, is now begrimed and black As mine own face."
I think Othello, in theater history, has been played as everything from an Arab to an African - depending on who wanted to play him and their own interpretation. Edwin Booth played him as a prince out of the seraglio while Paul Robson, obviously, played him as an African. And you’re right, many references to black and sooty skin within the play.
I like that sometimes in the opera, the singer will black his face but leave his neck and arms white!