Cleric's spells were always prayers. That's why they can cast them while wearing heavy armor. The RP reason against armor is it interferes with delicate hand movements.
Actually, the more accurate thing is that they're a conduit for divine power rather than directly manipulating it themselves.
Which is why Druids, Paladins and Clerics get the full spell list automatically, they don't have to learn to manipulate the power into a useful form. They just access the power and funnel it into the world.
Whether you RP this as them "Praying" to channel the divinity or some other method of calling their power is non-canonical (At least, I can't recall any references directly stating any Divine casters having to pray to a deity and have the deity cast things on their behalf)
If divine casting was based on praying to a deity, then Druids would make no sense. Since they don't necessarily follow a deity (Shadow Druids are a thing that exist, as well as other sects that don't follow say, Sylvanus) and if they did, they'd be Clerics not Druids...
That said, some Cleric abilities ARE based on prayer. Such as Divine Intervention, where you specifically call upon your deity to aid you (And it is a one time thing because ain't no-one demanding a god to attend to them more than once...)
