Parmandur
Book-Friend, he/him
PHB is clear about not all priests being clerics. Whether or not all clerics are priests depends on how you read and regard the description in PHB. I would say it does not say that clerics are priests and also it does not count as rules anyway. But stressing a different word would make it say that clerics are special priests.
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The accruement of being a priest in an organized religion comes from the Acolyte background; the Cleric is, counterintuitively, not clergy per se but can be seen a a prophet or divine Chosen One...