Strategy games proved long ago that genres don't have to literally describe their genre specifically. "Strategy" could conceivably describe Pong.
I'm pretty sure what people mean when they say "MOBA is vague" is "I dislike that a company created this term to market their game." But instead of actually saying that legitimate criticism, everyone needs to create contrived standards for proper genre names so they can dismiss it. I can't figure out why.
I guess my description of a strategy game wasn't a great one, since pretty much every game has strategy, or at the very least tactics, and I don't think that simply having the ability to form strategy is sufficient to be a strategy game. And the definition of a strategy game is heavily based off connotations that most of us have built up about the term. But I guess genres in general tend to be nebulously defined and people will argue about anything and everything on the Internet until the heat death of the universe.
Really, I guess part of what I dislike about the term "MOBA" is the "Arena" bit, since to me, the map size is excessively large for anything claiming to be an "Arena". But that's partly personal preference.
And on a side note, I wasn't even really aware that MOBA was invented by Riot until after I hated the term, and I'm more of a fan of Blizzard's "Hero Brawler". Or, in the spirit of roguelikes, DotA-likes, but good luck getting Riot to acknowledge the existence of that game.