Rise of the Argonauts sounds more like a dating sim to me.
>_>
Soul Calibur 3 had a couple play modes that verged on RPG-esque, one by your definition and one by mine.
The first one is Tales of Souls--single player, you pick your fighter, and then after each round you get to pick one of two possible paths, which determine which way the story goes. "Do I search for more information about the sword, or about the man who wielded it? Do I go east or west? Travel by boat or by land?" etc. Story changes, you fight a different opponent, get different bonus stages. No stats.
Yes, you can get really into your character's head and have fun with the story. You're really deciding the path it takes, figuring "What would Kilik do?", even though you know that in eight battles you're going to be at the same cathedral at the end and fighting your destined opponent, then Siegfried or Nightmare, then the big bad. (And if you're lucky, you won't run into Setsuka on the way). I can enjoy MORE "What would my character do?" in this, than I can in Final Fantasy games...because it's trivial, really, I know I can't hurt myself with a bad decision, it's all just for kicks. But I REALLY don't consider it an RPG, even if there are minor roleplaying/story elements. It's not a Fighting Game/RPG. It's just a Fighting Game.
The other one is Chronicles of the Sword. It's this weird Ogre Battle-like strategic thing, where you send out units along paths on this strategic map, and try to take castles etc. When your units meet the enemy, you can let them auto-fight, or you can step in and fight it out in a standard fighting game thing. The game has a story, but you have NO decision-making power and the story has no real connection to you (there's a couple times where you can save some guys from the enemy, but it doesn't change the plot meaningfully). The RPG-ish elements come in where your units gain levels by fighting battles, and have stats that are different based on their "class" (even if they are transparent to the user, really--just that higher levels are better in a fight). Your units persist through different maps, and you can even grind for exp by losing a battle over and over. This is KIND of a Strategy/Fighting/RPG (what a weird damn combination), right on the verge between that and Strategy/Fighting.
You'll notice that these game modes are generally billed in the same way I describe them--ToS is never called an RPG, but CotS often is. Kind of funny how ToS has better story and actually allows decision making and getting into your character's head, but CotS totally doesn't. Yes, usually RPGs are more about making strategic decisions and building characters over time.
Diablo is an RPG. The only time I ever found myself thinking as my CHARACTER, instead of as a PLAYER, is when I did weird challenge runs (like some lycanthropy-based thing in Diablo 1 where you have all these play restrictions and player-instituted quests, like having to fight bare-handed and weaponless until you die at which point you can go start learning magic as a way to 'cure' yourself, but you can still only use very specific gear)... In that case, I was definitely playing in a pre-defined role, not one I created myself, not even with my own rules (I used ones I found online), but it still felt like actual roleplaying, especially when you interacted with other players online. Note, again, the big difference between roleplaying and RPG. Only rarely do people actually roleplay in what is commonly described as an RPG. Hell, look at half the online RPGs out there, people don't roleplay on those, they just grind their characters--you have to look for specific roleplaying servers. If "MMORPG" implied roleplaying, for example, why are only some of the servers "roleplay" servers?