The different perspectives are confusing.. anyways here's my take.
There's a distinct difference between games, and houses.
First-party... In the same building as the host company, or very very close. I.E. EAD, NST
Second Party... A drive away, on another continent, whatever. Just not part of the main building. I.E. Rare, Retro, SK
Third Party... obtained rights to make games on the system.
That's why Factor 5 is still third party. No owning--well, none that we know of.
In terms of games..
First Party... Developed and Published by the host company
Second Party... Published, but developed by someone else
Third Party... Not published by the host company.
Well, there you have it. At least, I think that's how it goes.