I noticed in one game where I was very technologically and militarily dominant, I had gone to war and later made peace with the Drath. Given how overwhelmingly dominant I was, they were willing to give me pretty much anything. I took all their money, but didn't really want their planets (as I was just turtling and enjoying myself), and they were basically encapsulated by the Arcaen Empire. Since I didn't have control over the U.P. I just decided to take a few million of their influence points.
As a result, not only did their U.P. votes drop, but also their influence borders collapsed as well. Not completely but pretty significantly. I didn't really want to see them destroyed, so I declared war on the Arcaens, bloodied their lip, and took a bunch of their I.P. Then I gave the Drath the I.P. I had gotten from the Arcaens, and as expected, their influence borders expanded, to the point where their influence was flipping the Arcaen planets.
So, yes, influence points you get from diplomacy (or demands to make peace) will affect your influence borders on the map.
One thing I noticed as well was that my borders did not expand by much. It may have had to do with the fact that the I.P I took from the Drath were a smaller fraction of my total I.P.. If I remember correctly, I also had to wait a turn to see the influence changes.
*EDIT: I checked this again by doing the same procedure with a different empire and the influence borders did NOT change. This is contrary to what happened originally, where the borders DID change. Is this a bug?
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