You might be looking too hard or overthinking it maybe.
If it's day 6, the password to save him is just one single word that'll stand out during the final scene of his bad ending.
Day 7 is a bit harder since there are three passwords. Two of them lead to another bad end, but in that bad end you're given the true password that'll let you progress.
But no, nothing really changes. Just pay attention to the dialogue and certain words will stand out or be given attention to.
Normally by Dave since he'll kinda figure it out, but need the player to finish connecting the dots.