I don't really think they are though...?
Depending on who you mean by "people."
NB I fully understand what you're saying and how you're defining it. I just don't think that's how people, as in people on here (and apparently Phil Spencer) are defining "first party" vs "third party."
In common vernacular I have seen no regard is given to IP ownership, they're simply referring to the source of development funding/publishing. Thus Heavy Rain or Sunset Overdrive or The Wonderful 101 are commonly referred to as first party games/exclusives.
i believe that if you're discussing what is a first-party game, you're actually discussing the holder of the intellectual property. nobody is actually questioning the status of the developer when they're talking about first-party games. hence the current discussion on sunset overdrive. it does no one any good if they think sunset overdrive is a first-party game when it isn't, and sunset overdrive 2 appears as a multiplatform release. it's fine for list wars, but it's incorrect to state it as a first-party game since insomniac holds the copyright and the trademark.
the wonderful 101 is copyright of nintendo/platinum games, and trademark of nintendo. that's a first-party game developed by a third-party.
what i don't know is what happens in the split. that is actually the case with heavy rain. scee controls the copyright while quantic dream holds the trademark. i am not sure what that would indicate for the series moving forward- i think the copyright refers to the product that is heavy rain- which scee is in control of, while quantic dream is in control of the logo and other sorts of likeness of the game. i really would need clarification on this, but i think it means sony can't simply use all of heavy rain name/logo without quantic dream's permission, where nintendo would be free to do so with the wonderful 101. quantic dream might feasibly be able to shop the heavy rain name around to other places, but not the original product made in 2009/2010.
anyway in the case of sunset overdrive, development funds or not, insomniac holds the copyright and trademark to the game. there might be some exclusivity stuff in there for microsoft where the game can't appear on other platforms (or maybe can't for a long period of time), but microsoft is also unable to use the sunset overdrive name/characters/worlds without the developer's (insomniac's) permission. that's something you could do with a first-party game, something sunset overdrive is not.
people are free to believe whatever they want, if they believe development funds have a purpose in deciding what a game is to them, but that doesn't mean they're right in a real-world sense.