I think each ship has it's C.O with a rank of Captain or something similar, and the Admirals just choose their flagships. So Petrarch possibly had a Captain XYZ that commanded the ship, like giving specific orders "Beams 1 and 2, fire at will. Helm, full speed, heading 2-3-7" whereas Petrarch was responsible for planning, and during the combat he would be like "Captain, engage this Moloch corvette and bring us out of this Demon's range", leaving details to the C.O. He would also coordinate multiple ships in the Fleet ("Actium, destroy this Cain crusier. Carthage, engage the Beast").
Also, I think we should not mistake the naval Captain and fighter Captain. The naval Captain is usually a C.O. of a capship, whereas a fighter Captain is a veteran pilot (possibly squadron commander) - see Captain Loukakis (sp?). I think that the player is awarded the "fighter" Captain rank in the game.