The Complete Divine from D&D 3.5e gives us insight into how organised religion works in D&D campaigns (TCD 130-134):
The god is the head of the church, followed by most commonly high-level clerics which are above the Druids and Paladins that would worship Pelor. This is a multi-layered top-down bureaucracy with the citizens at the bottom.
In Global Churches:
A Council of highest ranking priests is at the top only superseded by the god, head priests for local temples follow in power. The church is organised in a network of temples, so clerics are often on the move and visit other temples.
The common citizens of the church are taxed by the church. They are at the bottom.
Such a church would act similar to a nation, but it doesn't have any land other than the temples. The highest ranking priests are almost all clerics. Some global churches have standing armies.
Both in Theocracies and in Global Churches, there will be factions that are in dispute over both secular and theological matters. Factions are called sects, and their conflicts are called schism. These sects and schism behave in a way that is suitable to that deity. So sects in the church of Pelor will not be corrupted and handle their conflicts open, most likely by trying to outperform their rivals with more acts of good.
In regional and dispersed churches:
A local high Priest, under the god, is often the local centre of power. While the structure is similar to that of global churches but every decision is for local matters. There is no network of temples with subordinates. The density of non-clerics is much higher, and overall the structure is less vertical, but the local high priest is still almost always a cleric. Dispersed churches of the same deity are commonly in conflict with other churches of the same deity, over-regionally.
In nations/worlds where churches are suppressed Cults may form:
A cult of Pelor would try to delivery acts of good while evading the notice of the (evil) suppressor. Most cults have often fewer than a hundred members with a single high priest at the head who is not necessarily a cleric. Cultists take precautions to act in secrecy and security in the form of codes, secret temples and other tools of espionage, even when they follow Pelor and would commonly act in the open.
Common lingo used throughout the recent editions of D&D, based on my experience, (2e and up) includes:
Worshippers (in order from least to most important) who consist of followers, acolytes, priests, head priests, high priest, the high council of priests and the deity/ies. There is much more specific terminology used in various places, but this (generic) language should apply universally.