In this case, none of this process was followed. To the best of my understanding, this is what happened:
1. Valve legal contacted Nintendo of America to ask "hey, what do you think about Dolphin?"
2. Nintendo replied to Valve "we think it's bad and also that it violates the DMCA anti-circumvention provisions" (note: nothing about violating copyright itself). Also "please take it down".
3. Valve legal takes it down and forwards NoA's reply to the Dolphin Foundation contact address.
This is very much *not* a section 512(c) takedown! Just standard legal removals / C&D between two companies.
This has some interesting and sad consequences: