Well for me personally, my requirements would be:
1. A RPG should allow for progression of character. Usually this is leveling up with stats, but it's not required that it be specifically that.
2. Have flexibility for role playing options out of combat as well as in combat(if applicable. Pretty much any RPG game has it, but I wouldn't dismiss the idea of a RPG without combat).
3. Provide choices that offer narrative or world reactivity
I'll admit to not playing JRPGs all that much, so I don't know how much I included those. I will say that something like Fire Emblem I consider to be a turn based strategy game with RPG elements rather than a pure RPG.
I'll also admit to being pretty flexible about it, since I don't actually care that much what a "real RPG" is =P
There is also the action RPG which honestly feels a bit like "It's sort of like a RPG but not really, so we're just going to throw the word action in front of it because we don't know what else to call it". In general these games still offer the progression of character but have limited out of combat role playing options and may skip out on narrative/world reactivity.
Diablo is the best example of an action RPG I can think of. You still level up your character and have stats with numerous options for ways to build your character for combat, but there's no real out of combat RPing and no choice/reaction in the game's story or world.
Since it's going to come up eventually, the only requirement of being a MMO is that you have a persistent online world where you're playing with hundreds or thousands of players at once. After that you either are a MMORPG, MMOARPG, or MMOwhateverelseyouwanttobe.
With this I would say that DA:I is a RPG because:
1. There are level ups with skill allocation, which provides the progression
2. Various dialogue options provide out of combat role playing options(even if limited to mostly good alignments), and the class/skills provide in-combat role customization
3. After 2 playthroughs I feel it does enough of this, but spoilers =P