In the Animated DC Universe, Batman is the first. In the first episodes of Superman: tAS, Martha Kent tells Clark she doesn't want him using a mask, or people would think he was like that horrible Gotham character.
BTW, the Barbara Gordon character isn't Batgirl/Oracle. It's James Gordon's wife (the Batgirl/Oracle Barbara Gordon is Gordon's niece/adopted daughter).
Gordon's only child is his son, named (iirc) John. In comics continuity, Gordon divorces from Barbara by the time he makes comissioner, and she moves back to Chicago with his son. Gordon then acts on his affair with fellow cop Sarah Essen (later Sarah Essen-Gordon). Sarah gets killed by the Joker at the ending of No Man's Land.