I guess my point was that every game requires someone to be the DM, and the DM is going to pick up the DMG so the information being in the DMG as a training tool would allow the DM to then communicate with the players and explain the details. If you put a Deity in the PHB, you are going to have a player who wants to follow that path, and if you don't allow it you are creating a disappointment (especially for people new to the game).
Far better for the player to read about the cleric, see all the class features, including domains (I'm just going to use 3.5E terms for simplicity) and see a description that includes a line reading "Each cleric will have two of the following domains, depending on their faith. Check with your Dungeon Master to find out what faiths are available in your game."
You can encourage the relationship between the DM as a world designer, and the player as an active participant in that world.
"Your choices are cool Dave, but I was really hoping to play a cleric who focused on growing flowers and dancing and had the Herbalizer and Entertainer domains... can we work something out?" (humor intentional).