It is a term used mostly by Death Eaters.
Interestingly, Harry himself asks Snape this very question, in one of their Occlumency lessons:
“You are lazy and sloppy, Potter, it is small wonder that the Dark Lord —”
“Can you tell me something, sir?” said Harry, firing up again. “Why do you call Voldemort the Dark Lord, I’ve only ever heard Death Eaters call him that —”
Snape opened his mouth in a snarl — and a woman screamed from somewhere outside the room.
— Order of the Phoenix, chapter 26 (Seen and Unforeseen)
However, they are cut off by Professor Trelawney’s screaming before Snape is able to answer.
Your theory that only certain people are entitled to use his actual name is backed up by a passage from Order of the Phoenix, when Harry is surrounded by Death Eaters and deigns to use Voldemort’s full name:
“How come Voldemort wants it?”
Several of the Death Eaters let out low hisses.
“You dare speak his name?” whispered Bellatrix.
“Yeah,” said Harry, maintaining his tight grip on the glass ball, expecting another attempt to bewitch it from him. “Yeah, I’ve got no problem saying Vol—”
“Shut your mouth!” Bellatrix shrieked. “You dare speak his name with your unworthy lips, you dare besmirch it with your half-blood’s tongue, you dare—”
— Order of the Phoenix, chapter 35 (Beyond the Veil)
It’s a subtle way for Voldemort to remind his Death Eaters that they sit below him, and enforce this fear. By forbidding his servants from using his name, they’re implicitly in a different, lower-ranking class. Only the worthy elite (i.e., him) are able to use his proper name.
The Death Eaters are as scared of Voldemort as anybody else; they just happen to be on his good side. They know he could kill or torture them if he got bored (witness Lucius Malfoy’s fall from grace). The same taboos around using the name in polite society may apply to Death Eaters.