Since there seems to be no relevant discussion to the actual topic, and no one seems to be bothered by this, let's have a go:
I prefer AoS because it tells me exactly what I want to know, at the cost of limiting the audience of people that'll understand instantly - it's a small team based game where players control only one unit out of a respectable number of choices from a 2.5D(3/4 persepective) camera view and face off against another team of players. The players job is to defend their base, which constantly pumps out weaker allied units while trying to destroy the enemy base, there's a dual/triple track of powering up through experience gained, gold to buy items, and sometimes hero customization in terms of skills (innate abilities, passives, actives, ultimates, whatever you want to call them). AoS is as pronounceable as "xkcd" to everyone I've met, that is unpronounceable unless you spell out the term, and sticks out in my memory more than MOBA.
MOBA sounds like a marketing term for a box rather than an actual genre, and doesn't explain much other than it's online, and it's multiplayer. So what? A lot games are online and multiplayer. If I was trying to sell the game to friends and told them it was a MOBA, I'd still have to explain the whole type to them, or explain it's not a FPS, a RTS, an action game before explaining what it is. For those that know what AoS is but not MOBA, I'd tell them it's AoS/DotA-like so I could get a funny look and be asked why I didn't just say that. For those that know MOBA, they probably know AoS, know MOBA and not AoS I could do my little spiel on the history of the game. It might be crafted to latch on to "MMO" for the unsuspecting consumer that has no idea about the type - which if they try it will probably end up in a lot of tears and flames without some handholding, which still doesn't have a proper tutorial yet. MOBA could refer to Team Fortress 2, or Mount & Blade: Warband, or Dawn of War II - these games are all multiplayer, online, do battle, and take place in an arena of sorts. MOBA doesn't have much mindshare yet, and AoS isn't quite corrupted in public image (DotA has it's stereotypical players) or broken, so why fix it?
And if we're naming things by first game, Maze Wars is a horrible name anyway. It's not as interesting as Aeon of Strife. But Aeon of Strife is a completely fan-made thing and seems to me to give a little credit to the fact that modders created it. It's similar to Tower Defense - Tower Defense existed as an idea and a game
earlier, but you could argue it became popular in Blizzard games, and it was there that the term "Tower Defense" took root. Of course, this could mean both of us are eventually ignored as DotA still has the largest number of people that have played it and relates as the first concept in their mind. The key difference between Tower Defense and AoS/DotA/MOBA is that the former can be distilled into a couple of words, the latter cannot. If not, why not use something interesting rather than generic?
Side note: For those future readers, the original idea was loosely sketched out in a the manual for Starcraft, and one player decided to expand on it with a simple map that's generally been credited as the beginning of the type, placing players against a never-ending horde that spawns from buildings, where the game is won by destroying the buildings that produce the units. The type eventually evolved into a co-operative and competitive version, the competitive version first really making a mark as DotA. Due to different servers and their populations other AoS-types eventually got pushed out and DotA became the king of the roost, even if the other maps were aimed at different niches of the market, think RTS vs TBS vs simulation wargaming. Other AoS types are probably still in development (haven't touched Warcraft III in a few months) and have attempted to push the edge of what's possible in this type of games for a while now, and if these games catch on we'll probably see prior work being done for at least 5-10 years as they cover old ground.