Hi, yes Lews Therin/Rand is a Hero of the Horn. From Robert Jordan;

"Is this soul born in any other Age, or only at the advent and (theoretically, of course) the closing of the Third Age, as the Dragon/the Dragon Reborn?"

ROBERT JORDAN

This soul is one of the Heroes, and bound to the Wheel, spun out as the Pattern wills. "It" is born in other Ages, but in a non-Dragon incarnation, to suit the pattern of that Age.

Remember that there is also a Female Dragon, in the person of Amaresu (a parallel of Amaterasu, the Japanese Goddess of the Sun). So we'll never know the "original" name of the Male Dragon...just that he was born as LTT in the 2nd Age, and Rand al'Thor in the 3rd Age. Perhaps the next time a "Dragon" is needed, it will be the female soul that is spun out...