Influence seems to be hidden stat and they change reactions (to me) more than mood. For instance both sisters can be in good or very good moods but depending on who has the the higher influence the homework and blog scenes change. If Eric's influence is higher (mood good or very good) Lisa homework dialogue changes to she has a tutor already and for Alice blog scene you actually see Eric there. If the MC's influence is higher Lisa's homework dialogue does not mention Eric and reverts to pre Eric helping her dialogue. For Alice it's splits her blog scene either states Eric is helping her or you get a scene where there is no interaction available. From what I'm seeing there may be other instances but I have not noticed them yet.
And all that amounts to diddly squat, I have 14 complete playthroughs in this update alone, in all 3 paths(no mod no need). All you need to do is complete all the opportunities that your path will allow, to be ready for next update. It's simple, if you want Eric, then let him have the influence, if not get rid of him, in the next update 0.07. Plain and simple, cut and dry.