Please bear in mind that this is entirely my opinion and I am in no way an authority on this subject. Also, I am almost certain that I am not the first person to hold this opinion or come to this conclusion. One final aside, I am mainly concentrating on the way it seems that most people only refer to the DotA-likes when using the term MOBA.
Anyway, I went to answer @Mike's poll about MOBAs and a thought occurred to me. First, I was going to answer that I have never played a MOBA before. I have the same problem that @jeff does with these DotA-likes; they take my least favorite part of an RTS and make it the entire game. That is where I stood initially, but as I said before, a thought occurred to me. Let's breakdown the acronym/initialization/whatever.
This descriptor is accurate though it does not hint that the player is controlling what is essentially an RTS unit.
Again, an accurate descriptor, but again it doesn't hint at the play style involved with the genre.
This is where I have the biggest issue. This descriptor is too vague. When you think about it, any online competitive shooter (CoD, Quake, Unreal) takes place in a Battle Arena. The maps that matches take place on fit the descriptor just as well as map(s) in Dota 2, LoL, HOTS, and other so called MOBAs. Sure, only Unreal and Quake are considered Arena Shooters, but the fact remains that even though the maps in other shooters aren't called arenas, that is essentially what they are. Merriam-Webster gives a definition of Arena as "an area of activity, interest, or competition." That describes maps in CoD pretty well. It cannot be argued that what take place on maps in CoD is not a battle, therefore those maps are Battle Arenas.
This fact is further demonstrated by fact that there are third-person shooter MOBAs. The only difference I see is that generally in MOBAs AI controlled entities are present and the player isn't necessarily just going for the most kills but none of this nuance is addressed by the acronym.
To be honest, I don't really have a problem with the acronym. It is in wide enough usage that most people understand what you are referring to, but I do feel it is far too vague considering none of the words the letters stand for address the specifics of the genre. Just my two cents.