The Dragon is the title for the male champion of the Light against the Dark One, and is always the reincarnation of a single particular soul. He is reborn in the Age of Legends and again in the Third Age (and equivalents along the Wheel) to combat the Shadow. It is unknown if he is born again in a similar repeating pattern in other Ages. The Dragon of the Age of Legends was Lews Therin Telamon. The Dragon of the Third Age is Rand al'Thor.
What Elan Morin Tedronai says in the Prologue of TEOTW indicates that Lews Therin Telamon only earned the name Dragon during the War of Power, so as Elan Morin Tedronai was named Betrayer of Hope. Lews Therin Telamon was never mentioned to be named Dragon Reborn.