Cleric was one of the original 3 as well.
They were one of the three classes in original D&D. However, clerics weren't in Chainmail.
They were added in one of Dave Arneson's proto-D&D campaigns when a player wanted a character that specialized in hunting vampires in order to oppose another player who was running a vampire as a PC.
The classifications of "fighting-man" and "magic-user" make a lot more sense in Chainmail, where it defines the role of the combatant and those were the only two roles in the game.
In D&D, the class names of "cleric" and - later - "thief", "paladin", "druid", "assassin", "ranger", etc. describe a profession rather than role. IMO, it has the effect of making the terms "fighting-man" and "magic-user" seem a bit awkward, especially since many of the other character classes also perform a fighting or magic-based role - but those two class names are just part of Chainmail's legacy that got carried into D&D.
