Again.
Core class == class in the core rules.
The core class are the Barbarian, Bard, Cleric, Druid, Fighter, Monk, Paladin, Ranger, Rogue, Sorcerer, Wizard; the Adept, Aristocrat, Commoner, Expert, Warrior; and the Arcane Archer, Arcane Trickster, Archmage, Assassin, Blackguard, Duelist, Dragon Disciple, Dwarven Defender, Eldritch Knight, Hierophant, Horizon Walker, Loremaster, Mystic Theurge, Red Wizard, Shadowdancer, Spellsword.
Base class: 20-level class a character can start in.
The base classes are Adept, Aristocrat, Barbarian, Bard, Cleric, Commoner, Druid, Expert, Favored Soul, Fighter, Healer, Hexblade, Marshal, Monk, Paladin, Psion, Psychic Warrior, Ranger, Rogue, Samurai (OA), Samurai (CW), Shaman, Shugenja, Sohei, Sorcerer, Swashbuckler, Warmage, Wizard. I would list also the core classes from Dragonlance (Mystic, maybe some other) and the Kalamar Player's Guide (Gladiator, Scout, Spellsinger, and some other), as both books have the Official D&D License stamp, but I don't have these resources handy.