All that matters is who publishes the game. If a first party publishes a game, its a first party game. If a third party publishes a game, its third party game.
A game developed by a third party, but published by a first party, is the very definition of a "second party" game. First party publishers approach third party developers to make games for them, likely pay for part of the development and marketing, and in turn get exclusive publishing rights to the IP, and likely some artistic oversight, so they can use the game as if it were first party developed. They'll focus on making the game more marketable and positioning its release to maximize console sales, not game sales, etc.
Its really not a matter of first party or third party games being more important. It doesn't matter who is making or publishing the games, if they are games with a high appeal. Right now, when Nintendo owns at least half of the 10 biggest franchises in the world (probably 6--Mario, Pokemon, Nintendogs, Brain Training, Animal Crossing and Wii-brand), and Sony and MS each own 1 more franchise in that group (Gran Turismo and Halo, duh), it seems that first party rules. But the PlayStation-era development environment for third parties is still in tact, despite the bulking-up of all three first parties. Some third party franchises will inevitably break through and become huge and defining games of the generation, like GTA, Kingdom Hearts, and even the likes of Need for Speed and Medal of Honor did last gen.
"[Our former customers] are unable to find software which they WANT to play."
"The way to solve this problem lies in how to communicate what kind of games [they CAN play]."
Satoru Iwata, Nintendo President. Only slightly paraphrased.
