Despite all of the members of this archetype lacking "Legendary Dragon" in their names, they each include a name condition that causes them to always be treated as "Legendary Dragon" cards.
The names of the "Legendary Dragons" come from the ancient philosopher Plato's workings. The specific body part in the name corresponds to where they were stabbed and sealed during the war in the Yu-Gi-Oh! anime.
"Legendary Dragon of White" is not a "Legendary Dragon" card because it does not have 「伝説の竜」 in its Japanese name. This is not an issue because it is not legal.
These cards are the dragon forms of the three "Legendary Knights" who used to live in the Dominion of the Beasts and answered Ironheart's call to defend Atlantis. Dartz transformed them into dragons to weaken them. The great ancient war ended in a draw, with the Dragons sealed away in their realm and the Leviathan severely weakened and diminished to a shadow.
Although their names are known prior to their release in the dub anime, in the original anime, no one—not even the monsters—knows the names of these dragons except for their chosen Duelists, who learn their name upon releasing them, without even being told what their names are. Thus, they are referred to as the Nameless Dragons in the Japanese anime sometimes even by their chosen Duelists.