1E - Character classes are restricted to Cleric, the Fighting man, and the Magic-User. Races are restricted to Human, Dwarf, Elf and Halfling. Alignment is reduced to Lawful, Neutral or Chaotic. Stats are rolled 3d6 in order. Further limit class choice depending on how low or how high particular stats are. Severely limit multiclassing.
While technically, you're correct, the standard nomeclature is Original Edition, or OE, sometimes 0E (zero-Ed)...
as 1E is usually used for AD&D 1E, the 3rd edition of D&D published.
And the OE classes with supplements include Cleric, Paladin, Thief, Monk, Assassin, and Druid.
The second D&D edition was Holmes, and it even predates AD&D, but you've made the classic confounding of Listing AD&D 2E as 2nd Ed; that is common but inaccurate.
Holmes basic classes are Cleric, Fighter, Thief, Wizard, Elf, Dwarf, Halfling. Elf (a fighter-wizard multiclass), Dwarf (a fighter), halfling is a fighter.
AD&D 1E is Assassin, Cleric, Druid, Fighter, Monk, Paladin, Ranger, Thief, and Wizard with Bard requiring dual-classing twice (fighter between 5-7 levels, into thief for 5-8 levels, thief into bard)...
Oh, and Mentzer, Moldvay are Cleric, Fighter, Thief, Wizard, Elf, Dwarf, Halfling, with high level characters allowed to transition Cleric-to-Druid, Fighter-to-Paladin, Fighter-to-Avenger. Cyclopedia adds "Mystic" (which is a monk). The Gaz line adds 3 prestige classes, "Ranger" (which is actually the same as elf, but is human), Dwarf Cleric (Fighter-cleric), Elf Treekeeper (Fighter-Mage with some clerical spells), Merchant ("secondary class" - equivalent to a 3E Prestige class), Sailor (secondary); adding the PC series adds about 100 additional racial classes; adding the HWR series adds another 30 or so.