Darth

The discription is a bit wrong:

The Dark Council officially recognized your authority as a Darth after you defeated Darth Baras (Sith Warrior) or Darth Thanaton (Sith Inquisitor)

Aye this one is for Sith Warrior when you complete act III of the SW storyline by killing Darth Baras.

Previously this gave you the title Emperor's Wrath. I don't know if this title is still in game but being the Emperors wrath is part of the SW story line so I'd think it's still there somewhere else they'd have to redo some of the dialoge in act III.

The whole renaming thing when you become darth comes from the rule of 2 in movie era lore. In TOR era were darth is merely a title of rank changing your name when you get it would be like changing your name in the military because you are now a captain.

Game mechanic wise, can you imagine the havoc it would have when you wonder why happy-bunny deleted his char and how the heck Frelinator ever got on your friends list

Not entirely true.

Revan and Malak seem to have kept their names when they went to the Darkside, so there is precedence

My question is whether Darth allows you to change your name specifically for the title or if it's just Darth <name>.

The latter would be disappointing, as no Darth has ever used their own name with their new Darth title to my knowledge. I mean, imagine, "Darth Anakin", that sucks. Likewise for "Darth Bob" or "Darth Joe", you should be able to make a completely unique name like Darth Malgus, Darth Zash, Darth Baras, Darth Vader, Darth Sidious, Darth Plaguis, you get the point.

If it isn't like this already, it should be.

Malak's real name is Alek, and Revan seems also to be a title [although no "real" name was given him for the book, so who knows?]. There are others who kept their name, however, especially in the million-and-one Sith eras.