As far as Im aware, all these Nintendo IPs are co-owned with Nintendo.So yes, they partly own the rights to said IPs and the definition stands.
About your second point: I think they kind of stop being first party.The whole point of calling something first or third party, a definition that we copied from other industries(as far as Im aware) is that first party is a term that describes that the platform holder of that platform is the developer of that software.The moment that dlstops being true, i.e. MS and Sony starts publishing their games in hardware they dont own/fully control, then whats the difference between calling them first party, or third party?
Whats the difference, regarding Ori on the Switch, compared to any other Square game for example?They both pay the same commission, they are both "aliens"(third party) in regards to that platform, and the companies have zero control over the platform.
The same applies to Sony games on PC.Why are we, in that situation, calling those games first party, if Sony games on PC has zero difference, in terms of definitions, compared to other third party games? From the PC hardware point of view, Sony games are as "alien" as games comming from Square, Capcom, Ubisoft, etc.
Thats the problem.People associate first party games as in belonging to the Big Three, and third party as the rest, instead of actually using their definitions to determine which term to use, because apparently thats too complicated.
The same happens with second party.I think its simple enough of a term, but apparently people dissagre with me.