If we were asked, I'd say give both sphere of influence(s) and national borders. Though national borders may require a tech to be unlocked or something?
what really me are the things we haven't seen yet, like inns. (your typo reminded me). We know we have dungeons, and national borders might cause trouble for that (because it would lock out certain dungeon access unless you had open borders... then players would get pissed when you walked onto 'their land' to get a dungeon)
Also with shardes and such. I mean, as much as I want to be able to say "My node! you can't have it" by putting a national border around it, I really don't want national borders to exist just so they are more contested. (in Civ, your boarders quickly are presseg against each other. No space of neutral zone where you are not sure who owns what)
If 2 nations can lay claim to the same tile, then there is no point in having boarders. because when I look at my screen I own everything I see, reguardless of what color they are. So if there exist nodes, inns, random pots of gold, fires (I'm not sure why you get so much random stuff from camp fires in HoMM) or whatever, and I can see them... I claim them. (unless somebody else beats me to there). Having a national border would change this... and having a national boarder that doesn't sync with what everybody else sees is useless
Scenario:
Player A: Hey, you grabbed that stuff. That was in my land!
Player B: Nah, my screen says my territory extends this far
Player A: But those bear cav units had to walk right past my city
Player B: Yeah, my map says its mine
(the city is player A's capital... and player B lies)