Logic dictates that if there is a first and third party, then their must be a second party.
I seem to recall first party, and third party, were existing terms. People then made up "second party" to try and describe organizational structures that didn't fit either category. In a conversation a while back, I believe the correct term for what people call "second party" developers is "subsidiary". Subsidiary companies may be partly owned by a parent such as Nintendo, or they may be fully independent.
What causes the confusion is that some companies that operate as subsidiaries to a larger publisher, appear to work with exclusive first-party resources and IP like Pokemon. Or they have a long history of making one particular series that's wholly owned by their patron. Those are really the result of exclusive contracts and licensing arrangements though - not necessarily an indication that the subsidiary is in any way owned (or even funded, outside the specific game project at hand) by their patron. So they're not "second party" in the way people tend to use that term.
A good example of a tight subsidiary partnership that is sometimes confuses the public is the development of Capcom's new gen fighting games. Some people frame them as "farmed out" as if Capcom had nothing to do with them, but that seems to be very inaccurate. For SFIV and MVC3, Dimps and Eighting respectively acted as subsidiaries to Capcom, with exclusive contracts. As subsidiaries they enjoyed the benefit of working with Capcom's internal resources - producers, designers, testers, and some content creators like musicians and composers. It's the kind of tight arrangement that Nintendo used for developing stuff like Metroid Prime.
By contrast, actual "farmed out" games happened when say, Capcom hired Grin to make Bionic Commando - there was nearly no cooperation or communication with the patron (and also no quality control).
As these things go, Nintendo has traditionally been pretty big on using subsidiary developers to create content or entire games. But compared to some companies, they usually (but not always) operate very closely with that subsidiary. Whether it's overseeing development with an unusual amount of micromanagement, or actually sharing internal resources and staff with the outside development house. I'm not sure how many subsidiaries are, in fact, outright owned by Nintendo, however.
By contrast, it seems (to me anyway) that the more common western method is to find a developer and buy them outright, then use them as just another internal development team that happens to come to work at a different address - with a strong tendency to destroy their internal culture and identity, and a nasty habit of treating the now owned developer as disposable temps as soon as a game is finished, absorbing or dissolving the former company.
This could be a reason why Nintendo works so closely with so many small developers but doesn't just buy them all to fold their expertise into Nintendo - to avoid destroying what makes them valuable in the first place.
