Both ARTS and MOBA are weak names for the genre.
But at the moment ARTS makes more sense then MOBA and therefore should be the official name.
MOBA means Multiplayer Online Battle Arena. Multiplayer and Online are two features of most games and only defined one genre, namely MMORPG because it was the key feature of that new genre.
Why would it be necessary to call those two features for LoL,HoN or DotA? We do not call Counter Strike a MOFPS right?
Then there is Battle Arena in MOBA. Those two words a pretty generic and could mean Angel Arena too (a mod for Warcraft III in which you would select a Hero and fight against other Heroes, sometimes with static creeps to slay for money). Those words could mean any game that has team-deathmatch.
So why ARTS?
It stands for Action Real Time Strategy. Let's start with RTS since it is an old well defined genre. It mainly contains ressourcemanagement combined with an amount of units that you would build and control. Furthermore RTS are played from topview. Ressourcemanagement and topview fit most ARTS/MOBA at the moment. Units not so much. There may be around 6-7 units you can control but those moments are rare.
Strategy is generic. CS has strategy too, no one can deny that. But DotA,HoN,LoL do contain several layers of that. It starts with picking, goes on with itembuilds, early movement strats and timings for important events (Rosh,Kongor,Baron) + roles of each player. Plus you need to adapt your strategy quickly to the enemies behavior.
So strategy fits in my opinion rather well.
And now the icing on top: Action. Most generic word there can be right? It accounts for the fact that RTS have a tendency to be rather unpersonal at the start because player need to build and send scouts first while in ARTS/MOBA player do have only one real unit that gets close and personal early on and need full attention. Making it a mexican standoff from the get go with high tension. Even zergrushes do not last long and can decide the game while in MOBA/DOTA there are towers to explicitly make an early win impossible. Juking is another point why the term action is right in my opinion. Every click must be thought through and enemies must be read.
I do think both terms are not ideally for the genre but i believe ARTS is a better one even if Riot managed to popularize MOBA as their marketing term.