Can you find a source for that? I've previously linked to Phil Spencer saying that first party game is anything they publish/fund.
Edit: Hell, Insomniac is even introduced as a "first party partner" at MS E3 2013.
(writing from my cellphone...)
Regarding the types of "true/full" exclusives, to make it simple, there are three types to which we categorize generally the exclusives...
The first-party type are projects usually owned by the publisher/platform holder and developed in-house by one of its own studios.
ex: God of War, Halo, The Legend of Zelda, etc.
The second-part type is a bit more complex as it could go in two different cases. The first, and the most common, is regarding how the platform holder will use a partner studio (i.e. Insomniac Games) to work on a new IP (Resistance for exemple) that will be fully owned by said publisher/platform holder. In the other case, you will have the platform holder work with a partner studio on an already established franchise/IP (Hyrule Warriors for exemple), which owned by said publisher. Also, it happens that the platform holder will treat those type of projects as first-party titles (marketing-wise) at major conventions or conferences.
For instance, before SCE acquired Media Molecule, LittleBigplanet could be considered a 2nd party title due to the type of relationship they (SCE+MM) had back then. Since the acquisition of MM in 2010, it is safe to say that LBP is a first-party title.
The third-party type used to be a thing back in the PS2/Xbox era but today it has become more of a rare thing to find these types of games. It is usually regarding a particular type of relationship between the platform holder and a 3rd party publisher (SEGA, Capcom, Square Enix, etc.) in which the platform holder (Playstation, Xbox, Nintendo) will usually help with the funding (and sometimes the development) of certain titles and help with most of the marketing for the other publisher (Monster Hunter, Gears of War (before the acquisition of the IP)). Those type of projects usually ends up exclusives despite the fact that the other publisher retains the rights on the said IP.
How would you categorize Insomniac Games' Sunset Overdrive then?
Personally, I would say that this particular game fits in... as more of a 2nd-party game. As they partenered with Microsoft Games studios to work on this title and the fact that MS helped funding this project. The only difference here is that IG still retains the rights on the IP on the contrary to usual 2nd-party titles.