If the players keep track of all the stats by imagining them, then they're playing an RPG "using" SS13 as their game board. That's what SS13 actually is, a big game board on which you can move your pieces around while you RP around it. SS13 keeps track of stats, by the way, just not all of them to make a complete character in itself. It tracks health, equipment, (dis)abilities and more, and those are central to the game's mechanics.
So, well, in a way I admit SS13 is an RPG if you take the entire experience beyond the software itself into account. It's sort of a medium for good ol' tabletop RPGs except online, and the stats are kept track off by the players using the role you get assigned or pick at the start. So you're correct, I did not really think about how the game is more than the software, so when the stats are outside the game the whole can still be an RPG.
As for Rise of the Argonauts, it's an action-RPG. The RPG elements are the traits, I read up about them and they indeed form some sort of character development. Even though this is very limited, current usage of the them RPG seems to allow it. It used to be that 'realtime combat' was unheard of in RPGs. And Jason is probably a very pre-defined character so it's not very imaginative RP.
Take away the traits/favors and noone will ever call it an RPG though.
Stats do indeed exist everywhere, but the whole point is not just stats, but stats that define the character. These also exist everywhere, everyone can be (approximately) defined by statistics, and every RP character can be put on a character sheet, listing all but perhaps its personality (besides maybe basic traits for that) and backstory.
If these kind of stats are not present in the game, the character is not present inside the game mechanics, it's only present in the player's imagination. This makes the game not an RPG as there is no RP *in* the game. The game could still be used to play an RPG game with, using the game as the world to play in, like SS13 or Second Life or even a forum RPG.
Recap: The character defining information has to be in the game and react with the game mechanics in order for the game to be an RPG. If they're outside the game, the game can only be an RPG if it is specifically designed as an aide for a larger RPG experience.