Throughout the series, Snape consistently refers to Voldemort as “The Dark Lord.” As far as I could tell, he never refers to He-Who-Must-Not-Be-Named as Voldemort, even in candid conversations with Dumbledore.
My assumption is that Death Eaters those with the Dark Mark are unable to ought not say the name, but I couldn't find any affirmation. Is there any evidence in-universe that explains this?
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.
Perhaps Snape knew about the Taboo curse, that Death Eaters could track someone if they said Voldemort's name.
This was only after Voldemort was revived in Goblet of Fire, but he could have learnt of this curse when he was part of Death Eaters, and thus made a rule never to speak his name when he was working as a double agent.
On http://harrypotter.wikia.com/wiki/Taboo (from Harry Potter and the Deathly Hallows)
Ron: "...the name's been jinxed, Harry, that's how they track people! Using his name breaks protective enchantments, it causes some kind of magical disturbance — it's how they found us in Tottenham Court Road!"
Harry: "Because we used his name?"
Ron: "Exactly! You've got to give them credit, it makes sense. It was only people who were serious about standing up to him, like Dumbledore, who even dared use it. Now they've put a Taboo on it, anyone who says it is trackable — quick-and-easy way to find Order members!"
— Ron Weasley to Harry Potter after learning of the Taboo curse.
The loss of Lily Potter was what prompted Snape to betray Voldemort. He lost the devotion that is so characteristic of all Death Eaters. But Snape had to protect his cover thus he used the title that all Death Eaters used to put Voldemort on a pedestal.
Snape also became fearless in the sense that for him he had lost what mattered most to him (this is seen in the Deathly Hallows when Voldemort is about to kill Snape, which he knows, but Harry sees that Snapes' eyes are fixed on Nagini and he does not attempt to save himself for fear of his life but only to somehow communicate to Harry what Dumbledore told him).
Thus Snape does not call Voldemort He-Who-Must-Not-Be-Named, as generally only those terrified of him are seen to do that, which Snape is definitely not!
Hope this answers your question!
Most of the death eaters called him the Dark Lord, including Snape, Bellatrix, and many others, as well as some non-Death Eaters, such as Ollivander. According to the Lexicon,
"As a term of respect, the Death Eaters would often refer to Lord Voldemort as "The Dark Lord." Voldemort demanded this, as making his name a forbidden word was a great way to instill a sense of insecurity and duty amongst his ranks. Essentially, he wanted them to worship him. This is why they would most commonly refer to him as "My Lord" or " The Dark Lord when conversing with him or speaking of him."
In fact, I can't find one single quote of a Death Eater calling him Voldemort, which makes me agree that the Death Eaters would be punished severely if they did call him " Voldemort."
The bolded part of the quote (emphasis mine) uses strong language that hints to a punishment if any of the times we've seen Voldemort dealing with disobedient followers is any indication.
An explanation for the non-death eaters that called him the Dark Lord is it is a term of respect, as most were afraid to say the name.
Because there is no canon mention of this, the best in-universe canon conclusion we can draw would be that Death Eaters faced some type of punishment for calling him Voldemort.
