I always believed that when it came to software for a platform the parties were referred to as such:
First-party: The owner/creator of the platform
Second-party: The user of the platform
Third-party: A developer who is not the first-party.
Now second-party seems to be used to refer to a developer owned/contracted by the owner/creator of the platform. When did this change in terminology come about and do we solely refer to "user created software" as such?
I agree, I always assumed it was a pun or extension of standard English grammar: first person, second person, and third person.
I think it went away around the same time the phrase "Second-World" disappeared.
I think the notion goes back to commercial arrangements. The first and second parties are in a direct producer-purchaser relationship. The third-party parties are those other than the first party that the second party is dealing with. (Note that grammatical person doesn't quite fit, but it is a cute idea. It works better if the first party is the customer and the second-party is the primary supplier.)
In this context, it is perhaps overloading too much on the term to say second-party developer. The second-party might be an IT organization and have to do many activities to install and use the products and services of the first-party. It might farm out a lot of that to third parties, too.
Perhaps the key thing is that the third party is generally not part of the (business) relationship between the first and second parties.
A first party developer is either the hardware owner, a sub division or a company bought up by the hardware company. A second party developer is a company comissioned to make a game from a concept by a publisher, while a third party developer is independent and work on games by using dev kits and licencing, paying a fee to the hard ware maker.