I'll correct the holes in your synopsis:

1000 years before the events of X, there was a war in Spira involving machina and summoners (anyone know why, or remember between whom?). A songstress named Lenne was among the summoners fighting against the machina. Her lover Shuyin refused to see her lay down her life, so he stole the biggest, baddest weapon from the other side called Vegnagun and used it against them.
The war was between Bevelle and Zanarkand. Bevelle had the advanced Machina of war and Zanarkand had Summoners as its defenders.

As for Shuyin, he was going to steal Vegnagun and use it against Bevelle, but Lenne stopped him from doing so. She didn't want him to kill anyone apparently. Subsequently, both of them were gunned down.


In the meantime, in order to fend off the machina users, Sin either appeared or was created (have the origins been accurately documented in either game?). Sin got out of control and laid waste to Zanarkand, and would have destroyed all of Spira had it not been for the summoner Yunalesca and her lover Zaon.
By the time Sin had come along, Zanarkand had basically conceded. Yu Yevon had the people of Zanarkand become the Fayth on Mt. Gagazet so that the Zanarkand they knew could be preserved forever. Their memories of Zanarkand were processed through their dreams, which he tapped into and gave form, creating Dream Zanarkand. Likewise, he created Sin from these dreams, only it was too much power for him to process and he lost control of Sin, even while he forced the Fayth to continue to dream, thus, he continued to summon Dream Zanarkand and Sin, and they couldn't turn it off.


Zaon became Yunalesca's Final Aeon, who joined with her to destroy Sin. Unbeknownst to them, however, any aeon who destroyed Sin would become Sin, reborn after a Calm of ten years, because Yu Yevon (the "God" of Spira) lurked inside Sin and would possess any aeon to enter its Sin-self.
The game never actually tells us how long a Calm lasts. It had just been 10 years since the last Calm began. That's all we know.


The Fayth, the spirits behind the power of the aeons, were obviously not pleased by their senseless destruction. They needed a way to defeat Yu Yevon itself, not just its armor of Sin.

So they fell into a dream, and their dreams created a memory of Zanarkand that was nonetheless somewhat real. People lived and breathed and were born and died in this Dream-Zanarkand.
As previously mentioned, it was Yu Yevon's idea that they dream, and he forced them to continue to do so once he lost control and became nothing more than an entity that lived for nothing but to summon.


Ten years before the events of FFX, a blitzball star named Jecht was training off the coast of his Dream-Zanarkand and encountered Sin, who also existed from time to time in this dreamworld (NOTE: anyone know how that would be possible?).
Both were made of the memories of the Fayth on Gagazet, so it would really only be like water flowing into water.


Instead of destroying this dream, which it could not do, Sin acted as kind of a gateway between Dream-Zanarkand and present-day Spira.
I just thought I should point out that time passes the same in both realms. It's been 1000 years in both places since Sin first came into existance. The two realms just lie in different places.


I'm going to check out that FAQ that ShivaBlizzard linked to and see if there's anything inaccurate about it. I trust very few FAQs from IGN, GameFAQs, or other places until I've examined them and confirmed their validity.