According to Michael Brooks: "Influence changes only apply to the involved parties and are only applied from combat missions or actions." (
https://forums.frontier.co.uk/showthread.php?t=97577&page=42&p=1691696&viewfull=1#post1691696) In other words, once two factions go into conflict they can only gain or lose points from each other.
Since Season 2 landed, this rule has been gradually relaxed. The current position is that the two factions can gain points from other factions and lose points to other factions - the conflict is no longer a closed sub-system. It's possible that the second criterion concerning the way point are generated for the two participants still applies.
What we definitely don't know - and FD has been asked about this - is whether this is a deliberate change or a bug.
This erratic behaviour explains why there is no definitive guide to the BGS, in spite of all out efforts.