"MOBA" doesn't feel like an appropriate term for the genre
I mean sure, there IS battling involved, and it is an online multiplayer game, but I would hardly call Summoner's Rift or Twisted Treeline "arenas." They're more of organic strategy game maps. Of the main maps, only Howling Abyss is even focused primarily on the combat/battle portion of the genre's moniker.
Dominion's Control points, and to a lesser extent Ascension's objective capping do fit a little better into the name, but they aren't primary game modes at all.
If I were to think of a game being called a "Battle Arena" I would think of something like Custom Robo, Power Stone, the vs Ape Escape game Pumped and Primed; games which are not fighting games, but do focus on combat in a competitive environment utilizing the features of the specific ARENA in their engagements.
Now obviously, the other, half-joking-half-not term for the genre is an Aeon-of-Strife-Styled Fortress Assault Game Going On Two Sides, used to describe the toxicity inhabiting a large part of these games player bases, but honestly, the term "MOBA" feels completely nondescriptive of the kind of gameplay a person outside of these games would expect of that name.
"Team Strategy" seems fitting, since the game is based on Warcraft, strategy and coordination play a key part, and that strategy is in the form of teamwork.
Just a random thought waiting for a potato at mcdonalds to get into my current game.