YuGiOh question: What is the story of the Destiny HEROes?

Yugioh Heros

FYI, the whole story with the "corruption by Dark Fusion and saved by Neos" thing is not really "canon" per se. Yu-Gi-Oh! card lores tend to be much more ambiguous, and the whole story of the Elemental Heroes as you've described is just the way some fans interpret the lore based on the card art, but that doesn't actually make it the actual story. Most of the "in-depth" lore stories are simply fan fiction written by fans, and are not official in any way.

Canonically, from what we know from the GX manga and anime, the Elemental HEROes are the monsters used by (regular) Judai, given to him by former pro-Duelist, Hibiki. They are mostly based off American/western superheroes.

On the other hand, Edo Phoenix plays with a Destiny HERO deck, which tend to play out more as anti-heroes than pure heroes like Judai's deck does. The Destiny HEROes are bound by the concept of fate/destiny and time, and have a British-theme. I suppose you could consider the Destiny HEROes as the "rivals" of the Elemental HEROes from a foreign land, who have different moral philosophies and way of doing things.

After Judai becomes evil as the Supreme King, he begins playing with Evil HEROes and Dark Fusion, kind of as a reflection of Judai's mental state at the time. Note that the Elemental HEROes were not necessarily corrupted by Dark Fusion itself, but rather by Judai's evil personality, which allowed the Elemental HEROes to fuse into Evil HEROes. Through the combined efforts of Judai's friends, they were able to defeat the Supreme King and turn Judai back to normal.

Finally, eventually Judai decides to create a HERO deck all of his own, since his Elemental HERO deck originally belonged to Hibiki, and Judai simply "inherited" it when Hibiki fell into a coma. The result is the Masked HEROes, which are based off of Japanese superheroes (i.e. a Kamen Rider-like influence).

Edit: The quote in my first paragraph wasn't meant to be an exact copy. It was just a paraphrase with emphasis (since Y!A can't produce italics), not a direct quote.