A Link to the Past was conceived as a distant prequel to the original Zelda, but featuring several common characters, thus being the genesis of the true iteration idea. The Link featured spends much of his time after this, his first game, travelling to other lands and so I have called him the Travelling Hero.
In this case, the designation of Hero is carried through the bloodline, similarly to Twilight Princess’s Link but dissimilarly to Wind Waker’s. How, if Timey died without descendants in this branch? Well, they both share the blood of the Knights of Hyrule, a group of people who in some way have been granted the blessing of the gods, even if there is no direct line. Other Links have earned their Hero status and the Triforce of Courage by proving themselves, but this one is born into his destiny.
After defeating Ganon and freeing Hyrule from its dark shadow (ie the Dark World), this Link has adventures mostly in other lands. I believe in relation to Link’s Awakening it is mentioned that Link is travelling to grow stronger, so that he can better defend Hyrule. Fair enough, as he is one of the last Knights of that bloodline. His first foreign adventure after LttP though is Oracle of Seasons, in which it is stated that the Triforce itself sends link to the land of Holodrum (this is anachronic, as it was developed after Link’s Awakening).
Therefore, I believe that the Triforce is initiating the training of this hero. Then again, the Triforce cannot distinguish good and evil. Perhaps, again, it is forcing the Hero to prove himself worthy of it. Whatever the case, Link journeys for some time (same hair, same shield, same Link!).
The fate of the Travelling Hero is not represented in any other game after this, but he pops up in cameo roles in other Nintendo games. This could be a part of his training journey before or after Link’s Awakening (sure, why not right?). This includes an appearance in Donkey Kong Country 2 and a mention in DKC3 (Earth/DK Isles), an appearance in Super Mario RPG (Mushroom World), and an unused sprite in Golden Sun 2 that totally counts because I say so (Weyard). Now these appearances are ambiguous and don’t have to be Travelly but since he is explicitly travelling to other regions, I put them here—also, these games were made after LttP and mostly before OoT. F-1 Race and Tetris are Decliney because they have trademark equipment to distinguish them (and were also made at that time). These cameos have the darker hair of that earlier design, but I felt better about putting them here anyway, for the reasons stated.
So that’s the “Era of Light and Dark”, according to Hyrule Hystoria. We don’t actually know what happened to Travelly after what I’ve outlined, or whether he ever got back to Hyrule. Things only got worse for the kingdom in the years following though, so perhaps he tragically never returned to the land he swore to defend. Or maybe he did, with the Triforce leading him home. I’ll leave that one to the fanfiction writers.