Both versions of Kirk did rewrite the simulation, yes. nuKirk programmed the computer to cause all of the Klingons' shields to drop - so yes, in that case, he gave himself the advantage. We actually have no canon idea what Kirk Prime did to the simulation. He could have programmed something completely different. All *he* said was that he made it possible to rescue the ship. He didn't say exactly how he did it, or what specific changes he made to the simulator.