I’ve worked on a couple of MMO’s, one a 3rd person RPG and another a sporting one (yes, really, and don’t ask it was rubbish!).
Just because a lot of people are playing a games does not mean that game is an MMO. Take the example of an FPS game from the original post, these are a “Multiplayer Online Games” but not “Massively Multiplayer Online Games” and the reason is simple.
With that FPS you load the game, get to the menu, open the lobby browser, set what gameplay type you’re looking for, find a server (or it matchmarkes or whatever) then you connect to that server with a fixed limit to the number of players in it. You play, you win/lose, you finish and you disconnect. So you weren’t online, you found a server and went online, you played online, you disconnected going offline again.
With an MMO you load the game, and go online into a central connected world where you character data is stored persistently online in that - and only that - online world. Once you’re there the online world might be broken up into pieces based on distance (for example) but at any time it’s possible for you to travel from where you are to anywhere else in that world and encounter other players without too many artificial boundaries. There’s no logging off and finding another server for example, you’re not limited to 32 or 64 other players, the number you see will fluctuate based only how many there happen to be within that particular area. When you logoff you character is (usually) no longer instanced in the world and it’s state, your equipment/health/details, are persistently stored in the games database. It is the authority regarding who you are and what you’ve got, not the slim copy which might be cached locally on your machine.
That’s the basic difference for an FPS or 3rd person type of game. Other game type blur the boundaries and allow for offline progression, but even then it’s usually a requirement to be online at the same time so that the data can be persisted/synced online.
Just my thoughts on it having worked on a couple and dealt with the server, gameplay and database tech’ behind them, it’s how we differentiated them and from speaking to others it’s how I think a lot of people would.