An emphasis on gameplay mechanics that would be consistent with tabletop RPGs, typically stat/level based progression, optimization of character traits/gear and choices in dialogue options/mission objectives.

Games that are not RPGs may include gameplay elements consistent with this (even if it's just a crafting system), but with greater emphasis on other genre-defining gameplay mechanics.

To me, turn-based combat. Nothing else is an RPG. And no, just because you have numbers flying out of the bad guys when you hit them doesn't make it an RPG. It's just an action game with stats. Let me against state that this is just MY personal definition, so please feel free to disagree. I guess I'm just old school like that, haha.

The line is blurring too much lately. I honestly hate how shooters are literally "tacking" on RPG elements, and being labeled as RPGs. I am sorry, but the Division is NOT an RPG.

Classes, the party/squad, stats, and quests are the major elements that define what an RPG is for me.

This is why its called a role playing game: Choice.
"choice" in character developement (growth).
Balance between choices, allowing various ways to accomplish goals.

There is no clear definition. Anything that has a level system has an rpg element. But then again so its for jumping or shooting in relation to shooters and platformers

And way too many rpgs have dropped them

It doesn't need to change as long as we uphold the original meaning ... 

It's only been largely forgotten ... 

In its very essence, a role-playing game is simply a game where the participants assume the roles of characters in a particular setting. Every aspect beyond that changes from game to game. With that said, the genre has strong roots to the pen-and-paper/tabletop RPG, and the aspects associated with those games have permeated within the genre throughout the years.

In modern gaming, those aspects (e.g., statistics, classes, experience, etc.) have spread outside of the genre, largely because the scale of games has increased. Gone are the days where a FPS was simply about shooting. There are FPS games now that include itemization/loot, experience/leveling, character building, and so on, and there are role-playing games that have included modern shooting or action combat. Hell, there are sports games with career modes that rival the most traditional RPGs in terms of "RPG elements".

I personally prefer the simple definition that a role-playing game is a game where the player not only assumes a role in a game, but has a certain degree of influence on playing the role (i.e., some combination of dialogue choice, character building, etc.).

Well RPG stands for ROLE PLAYING GAME (which everybody already knows) so it's basically saying you will be role playing something. The game has to make you feel like you're in that world, as if it's a second life you have. That's why stats and levels are in there to make you feel like you're getting stronger like you do in real life as well as adding more complexity to the game.
There has to be choices and repercussions from them choices (just like in real life) there has to be the feature to buy/loot or find new cothes/gear to wear just as you would in real life if you were in that situation. Good RPGs will also make you enjoy and like the characters it should make you feel for them and get a connection. Remember we're supposed to be Role Playing.