To me, an RPG is a game where you take the role of a character, and through your choices as that character, influence the world around you, even if it is minutely. It may be a personal story, but you are playing the role of a character.
For example, Harvest Moon, to me, is a slice-of-life RPG, with barely any story or long-term connection to characters. But it is a game where you take on the role of a farmer, and what you choose to do as that farmer, like the crops you grow, how you expand your farm, change the world around you, even though it's limited to just your farm and the relationships your character forms with the townsfolk.
A simple, less-is-more approach. And one that's not for everyone. And also one that most people don't consider an RPG. I barely do myself, but I qualify it as one on the caveat that it's not a action/adventure RPG as we've become accustomed to, with Baldur's Gate, Elder Scrolls, and even Dragon Age.
Jade Empire is a game I'd consider an action/adventure RPG, despite the fact that you can't create a character and you don't really manage new skills so much as decide if you want to invest xp in making some of your favorites more effective in a fight but otherwise the skill remains unchanged, and you only choose a pre-made character model. But you play the Role as the last Spirit Monk, and your choices change the world around you, from condemning Tien's Landing by keeping the river low and force them to survive without trade from the river, or by closing the Great Dam and allowing trade with the Empire to flow again, to the more radical choices that affect the course of the empire, even the option to decide that main bad guy is right and let him win.
The Mass Effect Trilogy is also an RPG, with classic RPG elements from a lot of great games, like a leveling system, loot, how to expand your skill set, as well as decisions that change the world/galaxy around you in a very real, direct way.
So to me, when I play the multiplayer, I'm not playing an RPG, I'm playing a multiplayer shooter with RPG elements in creating a character and managing how you level up and skills you choose, but that's about it. While SP to me is where the meat of the RPG is, playing the role of a character, and your decisions alter the world around you, and it's done with literal galactic consequences.
And I'm perfectly fine with that. When I'm playing SP, I want to role-play, and allow my character to become who I want them to be, or in the case of my Inquisitor, who's personality just evolved on my first playthrough and I ended up making choices I didn't think I would make if it were actually me because my Inquisitor seemed to develop a life of his own based on responses he made in prior conversations, but if I'm in MP, I'm perfectly fine simply doing my best to kill the enemy and bolster my allies, no need for story at all. It's about the competition.