ORIGINAL: forsaken1111
Let's say you discover the Loros fruit. It is a resource which is extremely valuable, but you do not yet know how to establish a long-term colony on a swamp planet. You could, and should, build a mining base there to harvest this resource, but if an enemy colony ship swoops in you have lost it unless you commit to military action. There is no negative impact on the other race for stealing this planet from you, your mining base is instantly dismantled (along with all of its harvested fruit!) which is a huge potential loss.
That is really the heart of the issue for me (at least the non sphere of influence part of it, that has already been flagged as an area for improvement) and is not entirely sensible. I don't think there is any need for an extra outpost system, a base would function fine as an outpost if the game actually recognised it as giving some claim over where the base is sited. Bases already have all the characteristics you want for an outpost except for that recognition.
Colonizing a world with a mining base that belongs to another empire is an aggressive action and should be regarded as such. I don't like it when other empires do it to me and it is silly and cheap that other empires don't care when I do it to them. There are several ways of implementing this effect which I think would all work well together:
Colonization targets with another empire's base on it should be covered by the "Allow colonization and mining stations in other empires systems" tickbox. For the AI it wouldn't require any additional code since they already do or don't do this based on the empire's characteristics;
Colonizing a planet with another empire's base on it should lower the "Our past relations have been" value for diplomacy purposes. Note that it should not incur a reputation hit of any kind;
Colony ships attempting attempting to colonize a planet with another empire's base on it should be regarded as hostile targets and the base and any ships in the area should open fire on it automatically and without any rep hit (as if they had been directly attacked by the colony ship). This is basically what players already do manually to prevent their bases being colonized but would require some subtle coding. First, something needs to be put in place to prevent the AI repeatedly wasting colony ships against a target (classifying it the same way that colonies with a pirate/life form detected warning are handled would probably work). Second, empires should take account of their relationship with the other empire before attempting it (for players it is a non issue since it is up to them how they value their relationships with AI empires. AI empires would be more likely to try and trade for the base with a free trade agreement/mutual defense pact while taking account of their current feelings towards the other empire and their own empire characteristics). Third, there is the matter of what that should do to the "Our past relations have been" value (probably okay to leave the current code in place which would lower this value if the first two points are addressed properly). Finally, there is the matter of armed colony ships (assuming the default colony ship designs will always be unarmed it is a non issue for AI empires, players should have to manually tell the colony ship to attack with associated rep hit just as if it was a separate military ship.);
Colony ships en route to a planet with another empire's base should be able to show up just as if it were a fleet on an attack vector, giving forces in the area time to rally (if their sensor network is able to detect it and they are able to spare the ships) and making it much harder to pull this off successfully against an opponent that is properly set up.
I think that would allow the AI behaviour to be as flexible and as close to player behaviour as is possible while giving players an interesting dilemma to face. Do you declare war, attack the mining base without declaring war, trade for the mining base or try to take it with a heavily shielded colony ship and suffer the relationship hit?
The only other thing to consider is troops. It would be nice to be able to board ships and bases with troops and take them over but there are a couple of issues that make it quite a complicated improvement to consider. As it can be treated as a separate issue it is a matter for another day and I do not think it is as much of a priority as making bases have some kind of claim over planets.
