I think many if not most online multiplayer games have become MMOs in the broad sense. To me MMORPG is a narrower concept, and seems to be the pretty much the same thing that Tipa means when she says MMO in this recent thought provoking post. I articulated "everything is a MMO, MMORPGs are different" much more fully in a post here a few years ago. However, I inadvertently typed 99% of a blog post in the response to Tipa's post, and I have never really explored where I feel the limits of MMORPGs here. So here we go!
To me what distinguishes a MMO from a MMORPG is not really any particular mechanics but a design sensibility. If the one of the central design goals is to create the illusion of a shared fantasy world (or fictional world in general), with at least some of the expected entertainment coming from social interactions with other avatars being run by players (even if indirect like an AH or looking at other player's outfits when you run through town) it's a MMORPG. It's more of a "I know it when I see it" than a hard limit, and any two observers might disagree on the edge cases. But if there is a straight line in the design history from the game to UO or EQ (or a MUD) I likely consider a game a MMORPG.
Social interaction is also key. The only way to experience the game has to be to hang out in at least some areas where other people you don't know could also choose to play. It's kind of like when you decide to leave your house and go to a store. You might not see anyone, but you certainly could and it isn't your choice. The only way to physically shop in the store is to take that risk. The only way to play a MMOPRG is to engage in shared social spaces that could, and in fact almost certainly do, have total strangers in them.
However that is far from enough, or Fortnight would be a MMORPG. Fortnight is definitely not a MMORPG to me, and the main reason is that design goal of the game is different. Some edge-case examples might give a better idea of where I am coming from.
To me DDO and Guild Wars 1 are edge cases, because the only areas that aren't instanced are villages, cities and keeps that lead either directly to quest instances or to wilderness areas that lead to more quest instances. However a major design goal is to create the illusion of a shared world inhabited both by yourself and other player characters, so I tend to include them. Diablo I-III don't count in my mind because you can get to the hub areas offline. It's by design that you don't have to fool with other players if you don't want to. Same with PSO, and any number of survival games like Arc Survival Evolved. However, a particular server of Arc that is always up and has a lot of players comes really close to being a MMORPG in my mind. Especially if it has more active players than can be logged on at once.
Second Life is another edge case, but I wouldn't include it because the original design goal is not to create the illusion any specific fictional world. The only reason I consider even an edge case that games made by users are now embedded in it, making a kind or proto-metaverse. Location based games are yet another edge case, but I don't include them because our world is so integral to them. I am on pretty shaky ground there with Orna or Magic Streets I will allow. However, I consider LBGs in general something new.
RPG is also a necessary part to me, but almost everything seems to have some RPG mechanics such as XP and levels these days, so presence of RPG mechanics is not all that useful by itself. But in what I think of as a MMORPG the mechanics are there at least in part to help make the world feel more fleshed out, kind of like in a PnP RPG. They aren't there purely to give players more things to grind towards.
All this is a very long winded way of saying I consider MMORPGs a specific genre with fuzzy edges. All bloggers and commentators seem to agree on what's in the middle, and there will probably never be a huge amount of agreement about where the exact edges are.
Whatever a MMOPRG is is though, it's a genre that's really special to me. They are a lot more enchanting and "sticky" than almost any other type of game. For me they have an indefinable magic that really makes them pop. It's a genre that hints at endless possibilities to me. However, it's a genre that also often makes me somewhat wistful, because I believe the potential will almost certainly never be fully realized by any design team. I'm not sure whether that's possible, even in theory, for a single game.
That said, I certainly love seeing how different teams of designers have taken a crack at it! Doing so most of my adult life has led to a series of fascinating journeys.