I think I'll just go with Sakurai's criteria for the time being.
Are they a character that originated in a video game? Have they been playable in a game on Nintendo systems? OK, I'll get used to it eventually.
I read the interviews. Sakurai didn't give too much of a crud about Nintendo relation when adding Cloud, but Cloud being playable on spinoff games on Nintendo systems actually did set his mind at ease when he otherwise would have been somewhat uncomfortable adding a character COMPLETELY foreign to Nintendo systems.
Yes, the spinoff games actually count. The criteria doesn't say WHAT game on Nintendo systems.
Sakurai never said they needed a Nintendo appearance. He did say it would make it more likely for them to appear, though. It's more like a "bonus" to him, but it's not a requirement. Being a video game character is a requirement. A 3rd party character was stated to require a "gaming history", although what that exactly means is hard to say. It seems to mean to be an icon, though. That said, whether it's been on Nintendo or not isn't entirely irrelevant, of course, but it's not held to that high of importance as people say it is. Likewise, these factors are specific to Brawl, Smash 4, and the Ballot. Melee possibly didn't have these standards, being he considered James Bond from Goldeneye. Of course, maybe these are why he didn't get in, but it could also be because of realistic guns, or copyright issues he doesn't want to deal with. Nobody can say for sure and he hasn't been all that specific much(I think an article did come up where he talked about copyright problems, but I can't remember. Push might know).
Of course, when it comes to a regular option between a Nintendo appearance or a Gaming Icon for a 3rd party character, the icon part would clearly be more important overall. Other stuff like how easy they are to get, and how to implement them are still major factors, although not strict criteria. Think of it this way; Bayonetta is a good example of why the ballot isn't about icons strictly. Under normal conditions, she wouldn't get in. However, she won the ballot, and was super easy to get. Let's say the next runner-up was Banjo Kazooie among 3rd parties. While he's more iconic overall of the two(although not a super iconic character, being he's not even on the same level as Rayman at this point, or even that close), he'd also be harder to get due to being owned by Microsoft. The fact he was on a Nintendo system meant little in this case. That doesn't heavily affect it much.
Let's take another example with Cloud; Cloud got in for being an icon. If he had zero Nintendo appearances, he still would've gotten in for how important he is to gaming rpg's. It wasn't a real factor overall, and one of the least important. Simply put, any gaming icon we get is going to get in as long as Sakurai can get the rights overall(this may even include Nintendo saying no because they don't want to support a character from another system, which is reasonable. This is more or less a factor against Master Chief, although not nearly as much as his weapon being a realistic gun, that is, his signature weapon). Promotional issues do matter for companies. They're still businesses. Keep in mind Cloud barely had any appearances on a Nintendo system at this point. No games particularly starring him. Just a few playable appearances at the time. Since Smash, he's gotten a bit more love on Nintendo, due to the warm reception of his playability. This also ties a bit into what I said. A character that appears in Smash may be wanted for a Nintendo appearance again. Thus, they have to be careful about it. They don't want to get demand for characters or games they can't secure easily for their system, meaning it could get fans riled up. While obviously Nintendo isn't the best at pleasing fans at times, they still have to think about many factors. What Sakurai wants isn't the only factor anyway. We saw how bad the Final Fantasy music lineup is in Smash. It's bad. Street Fighter is almost as bad. He can only do as much as he's allowed to do.
There probably will be other icons of interest to Sakurai to add, but whether or not they're on Nintendo already isn't a big factor here. It's whether they already started off a video game character and he can secure the rights without any issues overall. People need to get off the mindset that they need a Nintendo history. They
don't. Now, unfortunately many still don't understand this and speculate characters based upon having a Nintendo history despite this. To be fair, SourceGaming did originally mistranslate that bit, so many didn't know the facts. Unfortunately, not many care and concentrate more on the Nintendo part than the gaming history part. :/