Who is Dark Souls' forgotten God of War?

UPDATE SEPTEMBER 1, 2016:
The release of Dark Souls III has finally answered this question. That game has rendered the speculation below completely moot and invalid. However, this post will remain preserved for posterity.

There is considerable debate within the Dark Souls fan base regarding whether or not Knight Solaire of Astora may be the forgotten son of Lord Gwyn, and the God of War. The game contains many references to a forgotten God of War, who was the son of Lord Gwyn, the God of Sunlight. The primary source for this is the item description for the Ring of the Sun's Firstborn:

"Lord Gwyn's firstborn, who inherited the
sunlight, once wore this ancient ring.
Boosts the strength of miracles.
 
Lord Gwyn's firstborn was a god of war,
but his foolishness led to a loss of the
annals, and rescinding of his deific status.
Today, even his name is not known."

The game doesn't specify what the God of War did to be stripped from the annals, but the blunder cost him dearly. As a punishment, Gwyn and the other gods rescinded the God of War's diefic status and expelled him from Anor Londo.

But it didn't end there. Based on the content of the game itself, it appears that the gods also removed all references to him. This included removing or destroying any statues depicting him and redacting his name from records. Both his name and face are lost to history, as is his fate. Was he cast out of Anor Londo? Was he made mortal? Was he cast out of the world, entirely?

The gods made an effort to eradicate all records and traces of the God of War.
Statues of him were removed or destroyed all throughout Anor Londo and Lordran.

If the Warriors of Sunlight covenant (lead by the God of War) existed prior to the God of War's expulsion from Anor Londo, then it would also stand to reason that the gods disbanded the covenant. It's highly unlikely that gods would have permitted the covenant to continue to function, since its followers would be able to continue to teach of the existence of the God of War. These followers would likely have been forced to renounce the covenant or become an underground cult. The fact that the covenant still exists and has followers is likely due to the waning influence of the gods after Gwyn left to link the flames and the gods were forced to abandon Anor Londo.

The Sunlight Medal does offer one clue as to the fate of the God of War and his covenant. It suggests that the forgotten god still lives, and still watches over his followers:

"This faintly warm medal engraved with the
symbol of the Sun, is the ultimate honor,
awarded to those who summon the Warrior
of Sunlight and complete a goal.
 
The symbol represents Lord Gwyn's firstborn,
who lost his deity status and was expunged
from the annals. But the old God of War
still watches closely over his warriors."

In any case, we meet one NPC Warrior of Sunlight during the game: Solaire of Astora.

Solaire doesn't reveal much about himself. He is a Warrior of the Sunlight who has a strong reverence for the sun itself. In fact, he is on a possibly futile attempt to obtain his own "sun"...