The thing about that (which makes no sense) is, if Kirk is going to basically lie in his
report, and makes the assumption that Spock is also going to lie in his
report (because Kirk saved his ass), doesn't it make sense that Kirk would think to sit down with Spock and get their stories straight?
Then change the ship's records (remember when Kirk and Sarek watched the recording of Spocks death in ST Three?), changing the records would likely involve the ship's top computer expert, Spock.