ZoC should work as 1).
Within the scope of the game, i think a ZoC is defined as the space in which you can deploy starbases and military ships without asking anyone => therefore borders as in political borders.
In the ZoC of another empire, you should not be able to move in military ships, deploy starbases or even claim strategic resources or colonize worlds. In fact, even unarmed ships like colony ships and scouts should not be able to cross an empire's borders without permission.
And all this is a good thing since it brings more diplomatic options on the board, treaties like open borders for civilians ships (means unarmed), trading rights, mining rights, colonization rights, grant military access, establish military starbase and maybe more.
ZoC should be defined by colonies only.
Influence sphere, as ParagonRenegade said, should be separate and should represent how much Coca and chewing-gum guys on the other side of the political border drink and chew.
Spreading influence maybe a problem if as said before, influence starbases can only be build within ZoC but it makes sense, since i don't think ex-USSR would have authorized USA to open a USA Influence Consortium in Moscow or USA authorized ex-USSR to open a Communism Propaganda Center in Washington.
I think that before making an agressive influence conquest, trade and good diplomatic relationships must be established since with good diplomatic relationships, trade flourish and with trade comes influence. Good relationships can enable the creation of a influence starbase in the other empire's ZoC, therefore cultural conquest.
Espionnage could be used to spread influence as well, with backroom talks and similar subversive operations.
With territorial proximity, influence provided by an influence starbase could just cross the political border.
However, i must say that i don't think colonies should just switch from on side to another. It doesn't work like that in real life. Colonies should switch only if claimed by an rival empire (which is most likely a friend since it has influence based on trade, good relationships, see above) or if they ask for switching sides and are accepted by the rival empire.
But a colony switching sides should always ends up with a great diplomacy penalty (of course the losing empire isn't going to be happy to see its citizens tax leaving).
Reading this might make you think that military conquest would be easier and therefore preferred by players, which is not a good thing because having different, equals ways of conquest is a good thing.
Well i agree, but i don't think influence conquest should be simplified, it's military conquest that should be harder, especially in the field of keeping the military conquered colonies under control: there should be riots, sabotages and generally speaking, unwillingness to obey the new ruler.
It should translate in long-time maluses in taxes, production and research provided by the colony slowly decreasing over time (maluses even higher than 100%, meaning that an angry colony should costs to maintain instead of providing anything). A revolution should even bring the colony back to the original owner if there is not enough military forces to keep the system under check.
well i'll stop here since i probably went a bit far away from the question and that the game is most likely too much developed to include all this but anyways, since i write all this, i should as well post it!