How is it a media buzzword, ruby? This is the way I've always thought of it- a 1st-party is the console maker and all the companies it owns entirely (100%). That would be like EAD to Nintendo or Sonic to Sega before they dropped consoles. A 3rd party is a company that is not owned by any console maker and is generally independant (some are owned by other 3rd parties)- this would include companies like Capcom, Namco, Sega, etc. A 2nd party is a company that is owned in part by the console maker- this would be like Silicon Knights or HAL to Nintendo. So in that sense, neither Factor 5 or Camelot are owned at all by Nintendo, making them technically 3rd parties. However, they act like a 1st or 2nd party in that they only develop games for Nintendo's consoles.
I guess, though, I could see how you could simplify it and include all the companies owned at all by the console maker into the 1st party category and all the potentially multiplatform developers into the 3rd party category.