How I understood first, second and third party is back from the day in the industry.
First party, studio and ip owned by company.
Second party is lose term, even first can be weird.
Third party, other parties.
But that isn't correct anymore, these days we also have publishing, and shifting allegainces and ownership which makes it quite weird.
For example in which category does Kirby fall? First, second party?
en.m.wikipedia.org
While independent, it has been closely tied with Nintendo throughout its history, and is often referred to as a second-party developer for the company.
Okay weird, pokemon is first party right? Also not, they are now jointly owned but started with game freak
en.m.wikipedia.org
Although these IPs and companies might already fully owned by Nintendo now.
So appearance for consumers and relationship between both companies does play a role. Calling pokemon and kirby second party comes over as trolling yet technically it was correct at some point in time.
That said things are ever changing. Yakuza (sega) franchise although owned ny sega was highly coupled with Playstation until a few years ago.
Then another question is Halo first party, developed by bungie before they were bought. IP was transfered when they split up from Microsoft.
Another example is gears of war, is it first party? Develop by the now famous creator of unreal enginer and fortnite. Epic games who existed long before those games and have transferred the IP to Microsoft when they parted ways.
Both studios left Microsoft and those games build by them created the legacy. If you only own IP but no longer the studios to produce those games is it still a first party title? Or has it become a first party title?
Mass effect had the same trajectory as Halo, Gears of war. And it could have been become first party.
So a title could be first, second interchangeable only looking back can it be clearly classified as.
To make my point even clearer would anyone call forza Horizon a second party game? IP owned by Microsoft build by Playground games independent studio, they have only been acquired years later in 2018 or something. But I don't think anyone would call that game second party.
Then there is the consumer perception which plays another big role, when the niantic pokemon game did well Nintendo stock soared up, until the investors realised that pokemon wasnt owned by Nintendo only joint.
Metal Gear, resident evil are perceived by many to be playstation games.
There are so many different example of game IP that we could discuss of which the definition in any of those categories is fluid. Ori, heavy rain, bayonetta, mass effect.
So this discussion will never have the correct answer as only time will reveal the full story of the IP.
