Is Gears of War a first-party game? Yes, Microsoft published it.
Is Gears of War a first-party game? No, Microsoft does not own the intellectual property (or did not at the time).
I don't see why either cannot be taken as accurate.
gears of war was a game that was owned by a third-party and published by a first-party. the term first-party refers to the console or system manufacturer. any other party to make a game for that platform becomes a third-party developer, as they are not owned by the first-party. when microsoft had rare make games for the gba and ds, that was microsoft making third party games on the ds. when they did the same, but for a nintendo ip, those games were first-party titles. when nintendo published professor layton outside of japan, professor layton was a third-party game developed by level-5. the game did not become a 'nintendo game' the same way luigi's mansion 2 is a 'nintendo game' because they actually don't own the rights to the franchise.
if you want to be more accurate, you can use the term 'first-party published games' to refer to games published by microsoft; you can use the term 'first-party game' to refer to any game created for a microsoft-owned ip and released on their own platform; you can also use 'third-party game' to refer to any game published for a microsoft platform that is not a microsoft ip. obviously there is an overlap between 'first-party published game' and 'third-party game' such as the case of gears of war. both are accurate. what is inaccurate is calling gears of war (the first four at least) 'first-party games', attributing to them properties that don't exist solely because of their publisher status.
if you want to drop 'publisher' from that first example, that's where there's a lot of confusion, and a very inaccurate representation of the situation.
and just so we're clear, the world is nuance. that's the point in having to explain these things in the first place. you're free to use colloquial terms for whatever reason you want, but if you actually want detail? you're going to need to accept that there's a lot more going on than a simple explanation.