RPGs as all games are defined by their mechanics or rules. In this day and age gamers seem increasingly blinded by the surface of games such as their graphics, story (and the choices that may exist within), voicing acting and other things that serve as backdrop to one's main engagement with the game.
An rpg is defined by the stats of the character or characters one is controlling and the overlying system that determines the effectiveness of action based one or more characters stats. Turn-based (and phased-based if it need be mentioned) rpgs are completely focused on the characters abilities. The player just chooses how to develop the character(s) and what abilities they'll use. The reason one would focus on a mage's magic modifier(intellect/wisdom/mentality) rather than strength, is dependent on the fact that their magical abilities won't improve and that that class generally doesn't have aptitude(stats&abilities) in physical combat. Now however, a mage-fighter would benefit from some increased stats in strength; there's the expectation that one will balance out stat increases in both magic and strength for that character while not doing so with a pure mage since it would be a waste.
Turn-based RPG Series: X-Com, Wizardry, Realms of Arkania, Metal Max, Chrono Trigger/Cross, Etrian Odyssey
Action-rpgs simply, or generally, remove the stats governing accuracy or hit and dodge. They typically only focus in on one character being controlled, allowing the A.I. to control the actions of one's other party members. Ultimately however, one's performance is dependent on the stats of one's character. After a certain point, due to scaling, one's character can't take on greater foes without increasing one's stats; this of course neglecting abuses to the system.
Action RPG (Series): The Witcher 2, Dead Island, Mount&Blade, Star Ocean, Last Story, Parasite Eve 2
The word role in RPG has nothing to do with story, but the abilities of one's character.
Rogue is a role in an rpg.
Cleric is a role in an rpg.
Farmer is a role in an rpg.
Miles Standard is a rather unassuming gentleman from the town of Elbe in Ugar, located on the Felkin continent of Sume. At the age of ten he was enrolled in the Magic Academy of Dindom, where he learned of the intricacies of magic. At the age of fourteen, a game of oneupsmanship with his colleagues resulted in him being blinded in one eye; in actually he gained the ability to see the little viewed world of phantasms. At the age 18 he found himself drafted into a war between the Principality of Ugar and the Holy Sumen Republic. At the age of 30, he was released from captivity and made a general in the Holy Sumen Republic due to his masterful commanding when he rose to the rank of lesser knight at age twenty-five and served as such up to his capture at age 28 at Azul Creek. Now in charge of his life, he returns home in order to decide what to do with his life.
None of that is a role in an rpg, just the founding of the player's starting point. One knows Miles has some experience in magic, but one doesn't know anything important to gameplay. Lesser Knight and General are just titles that give one no idea of how his skills have developed.
Choice is ultimatly insignificant compared to the purpose of the start and endpoints in an rpg. Multiple paths and faction allegiances serve to limit or open one's options. However, they will eventually submit to the end. Those games that could be played indefinitely end when the player says it ends. The path be it straight or webbed functions the same as levels, tracks, rooms, areas and stages in other games. The ability to either use magic, a sword or a gun to kill enemies also falls into this category dubiously and only then on the account of using more than one option to solve problems is widespread through video games. In Smuggler's Run, one's choice of vehicle can make difference between easy success, hard success, neck and neck failure and guaranteed failure.
Leveling are insignificant compared to stats. Levels simply allow one to improve a character's abilities and stats, or just increase a character's repertoire of abilities. A great example of stats being more important than leveling is Trapped Dead. The reason you would give one character a gun and another a melee weapon is dependent on their stats. Take the doctor, it only makes sense to give him the shotgun, do to his low accuracy. Giving the bandages to anybody but him is lunacy, given the fact that he is the only one that can heal characters. Everybody else except for the photographer I think, just stop the bleeding.
I purpose this simple question. What is Dungeons&Dragons without the stats and D20 system, but keeping the lore?
To me, it's a just a collection of information and a bestiary. One could say that without the dice rolls and character sheets, one just has the components for creating a story. One could say its a Create-A-Your-Own-Choose-Your-Own-Adventure-Kit.
The idea that one doesn't play rpgs for the gameplay is absurd. Story is context, it explains the how and why. Story is the alpha and omega.
Zelda is an action adventure game, the genre that relies entirely on player skill and none of what makes Link. To the adventure genre entirely, puzzles are their most important aspect or the concept of how this puzzle prevents one's progression into the next room. Action adventure simply adds constant and ideally competent combat. Link's items, be they weapons or tools, are no different from the pistol and gravity gun of Half-life. The idea of comparing Zelda to rpgs stems from the fact that it has few competitors in the genre; especially not one's that are re-occurring. Heck, the whole reason that one saw people constantly comparing Skyward Sword and Skyrim is because they both lack what one would call competition; the contest between rivals of attempting to redefine or improve the basics of their genre.
In short:
Systems and stats define rpgs, player choice and character advancement are marginally important in determining what an rpg is.