I actually don't really care about the naming conventions per se (whether they use cleric or priest, wizard or mage, rogue or thief, fighter or... whatever)... but I DO think it's important to have "generic" class builds, as well the optional extended builds with things like domains, schools, weapon specializations, and expert fields.
At the very basic foundation of the system, for those people who didn't want to deal with all the optional "character building" stuff from 3E and 4E and instead just wanted to have a "Fighter", or "Cleric", or "Wizard", or "Thief" a la Basic D&D... it should be possible to create within the game these basic "iconic" builds of these four classes. Nothing fancy, nothing much different than what the four classes would have gotten at each level in BD&D. "Generic" fighters, clerics, wizards, and thieves. Pick one and go!
Conversely... I definitely believe there should also then be expandable systems in place to build out those four classes so that they CAN all be different from each other. So that a "cleric of the god of the sun" has different abilities and spells than a "cleric of the god of storms".
And whether or not you want to keep calling these expanded classes "clerics", or start calling them "priests" just to differentiate them from the basic version... doesn't really matter in the long run.
Personally... I would call them the different names. The Fighter is a generalist combatant who is good with weapons... the Warrior focuses on one or two special weapons, doing all manner of interesting maneuvers and exploits with them. The Cleric is a divine agent of the gods, spreading faith to the flocks while wearing armor and wielding a mace. The Priests each follow the tenets of a specific deity, gaining abilities that go along the domains and portfolios of the deity they worship. The Thief is a general adventurer and dungeoneer, skilled in your basic adventuring skills of locks, traps, and climbing walls. The Rogue is a specialist in any manner of adventure-- master cat burglar, diplomat, sword-swinging swashbuckler, assassin, pirate, cutpurse. And the Wizard is your generalist spellcaster, whereas the Mages all have a particular school of magic they specialize in-- the Illusionist mage, the Evoker mage, the Conjurer mage etc.
At the end of the day, it really doesn't matter if the basic, easy-to-learn classes have the same or different names as the "built" ones... it just comes down to what's easiest for new and experienced players to understand.