Almost right Demosthenes, except for you second party definition
Second party is when a company colaborates with the console maker to make the game. A lot of second party companies are partially owned by the console maker (Rare is partially owned by Nintendo). However, just because the developer is second party, doesn't mean they won't make games for other companies. A good example is Lucasarts. Shadows of the Empire for the N64 was second party, but they still make games for other systems.