third party is a developer who makes games for a first-party developer's hardware. 2nd party is a term nintendo came up with to describe third-parties who worked exclusively with them during the n64 era.
all that really matter is this- if the developer is a third-party and they're making a first-party property- you can treat it as though it's a first-party game (see also, the psp god of war games, luigi's mansion 2, and bungie with halo odst and reach). if it's a third-party making an exclusive title they hold the rights to, you can treat it as a third-party game, although the likelihood of it appearing on other systems can range from unlikely to nearly impossible (see also bayonetta 2, danganronpa, and dead rising 3).