So, more and more people use MOBA, and it looks like game publishers, game companies and review websites start using the name, so I am probably too late with this. But better late than never.
MOBA, short for "Multiplayer Online Battle Arena", is a term often used to describe DotA-like games, which only recently have come to be called an own genre.
But let us think what the single elements of an acronym of a game tell us.
Let's start with something else than MOBA so we get a feeling for it.
RTS (Real Time Strategy):
Real Time says that it is not turn-based. You order your units and/or buildings around, their supposed reaction occurs immediately, this can happen parallel to someone else's orders.
Strategy says that you don't let your units walk around to water flowers, but actually focus on strategic stuff - which usually means placing your troops and fighting.
This is a pretty accurate description of what you're doing in games like Starcraft 2 or Warcraft 3, and it would hardly fit to other games like Counter Strike or RPGs (both got a stronger focus on other things than strategy).
FPS (First Person Shooter):
First Person simply shows the perspective in which you are playing - you see from the eyes of the character you play.
Shooter shows the focus on, well, shooting things. Don't expect to run around and do highly complex stuff, shooting is the central element.
Again a perfectly fine description for the games that are entitled FPS (for example Battlefield, Counter Strike), anyone calling WC3 an FPS clearly failed to understand the system.
And now take a look at MOBA (Multiplayer Online Battle Arena)
Multiplayer Online is just a generic "prefix", we also got this in MMORPG (Massively Multiplayer Online Roleplaying Game, like World of Warcraft or Eve Online), so in general there is nothing wrong with it and it doesn't say much.
Just like in MMORPG, we can ignore this prefix and are left with:
And this is where I fail to understand how "Battle Arena" describes DotA-like games in any way.
Battle Arena can be SC2, you have a certain map ("Arena") and battle in it.
It can be a Beat em up like Tekken, you have limited area ("Arena") and battle in it.
In fact, it can be any game where you fight and have an area fitting to the definition of the word "Arena" as it is used in "MOBA".
Feel free to use this, but please give credit.