I've sent a few mod notices over the past few months about using the Nintendo tag, specifically about adhering to Danbooru's rule regarding its usage on posts that do not feature references to the company or its hardware/employees. However, Danbooru's rule recently came to the attention of lozertuser, who completely disagreed with it. As such, the Nintendo tag should now be used on any first or scond party Nintendo games. If I've sent you a mod notice about this in the past, consider it null and void.

tl;dr: the Nintendo tag can now be used on posts that simply feature Nintendo characters.

Just to clarify, is this a comprehensive list of "second party" developers?

nintendo.wikia.com/wiki/Category:Second_party_developers

It might be, I have no idea. The answer probably depends on the reliability of that Nintendo wiki.

Seeing as how second party developers are able to move between companies after their contract with one company is up, the tag shouldn't be added based on the developer of a game but rather the game itself. I'd say that if the game was exclusively released on Nintendo consoles, was made specifically for Nintendo or is iconic to Nintendo for some reason then the tag applies.