While I agree with you, it should be noted that this depends entirely on the definition of MOBA that one chooses to use, because due to developments in how games nowadays are mostly consumed in a multi-player fashion, the term became somewhat ambiguous.
CLASSIC Definition
In its original or classic definition, MOBA is a very well defined genre, with games like League or DotA. In that definition of the term, most FPS games are not MOBAs, with the notable exception of games like Smite, which have elements of FPS in the way they are controlled and presented, but function according to similar overall rules and setup.
NEW Definition
In its new, or extended definition, which is the one you and me are using here in this thread, the term defines all team oriented multiplayer games, where certain goals have to be achieved by one team of players against another team of players.
This includes games like CS:GO,Overwatch, League, DotA, MechWarrior Online or Dreadnaught.
So MOBA is less a genre nowadays, its more a definition of what the player can generally expect the game to be about:
- They are PvP and (as a logical consequence) multiplayer focused
- They are focused on online-content and gameplay
- They are highly focused on team-based play
- The game runs mainly or exclusively in the form of instanced "matches"
- They include some form of automated or semi-automated matchmaking
Whats NOT in this definition are games which focus on the players themselves to make the matchmaking (like shooters worked in ye olde times without publisher provided servers), MMOs (while mobas have "massive"ly big playerbases the number of players in each instanced games is not "massive") and competitive PvP games which focus on 1v1 content such as Starcraft or Hearthstone.
Whew, that post grew longer than I expected :D
As a closing remark, the "modern" definition of the Genre into which games like League and DotA belong would be: action real-time strategy or ARTS for short.
[slayer-jinx-catface]
######TL;DR: MOBA used to describe a genre, now its a term describing a more general description of what a game is about. The actual genre of games like League is nowadays called A.R.T.S.