Yes, they are. Have you ever seen a legal document? Essentially a first party is any party involved in the contractual agreement, and a third party is anyone who isn't.
When people refer to first and third parties, they refer to the publishing rights, not the IP or product rights. Atlus, or Namco, or Sony depending on region, two of which are third party publishers, one is not. Moreover, I don't see how that changes anything if you were to reintroduce the second party term, it would still be second party in Japan, and third party elsewhere.
Second party is hazy and murky because it's bullshit that does not exist, unlike the other two terms which are in basically every publishing contract these companies produce.
And yes, Mass Effect was first party, until it was third party, and no, that's not ridiculous, it's factually, contractually true.