I would say a third party is a separate independent entity, and a second party is a separate but owned or semi-owned entity. Hence a third party game is one made by an independent company, a first party game is made by the console manufacturer, and a second party game is made by a second party company, one that is a separate company but largely owned by the first party.
A game I'm developing with some friends:
www.xnagg.com/zombieasteroids/publish.htm
It is largely a technical exercise but feedback is appreciated.