The ARTS name predates the MOBA name iirc, but the ARTS term was unpopular because it's plainly obvious that they aren't RTS games and they were only called that because they fell out of the modding tools for Blizzard's RTS games. Ironically the existing genre they resemble most is ARPGs since they're basically ARPG gameplay and progression mechanics in a team vs team setting, but people get grumpy when you bolt the term "RPG" onto something with no plot to speak of.