(SPOILER)
Quote Originally Posted by Datalog: History and Society: Pulse Fal'Cie
Fal'Cie are an existence beyond human comprehension, possessed of incredible magic power. The ones responsible for Cocoon's construction are protectors of humanity, but there are others of their kind as well: the fal'Cie who dwell on Pulse and name themselves enemies of Cocoon.
Humans who encounter Pulse fal'Cie are cursed, being turned to l'Cie and ordered to destroy Cocoon. It is for this reason that most ordinary citizens support the Purge - anyone who may have come into contact with one of these fal'Cie represents a dire threat.


I don't know if Lindzei actually existed or whether it was a lie that the Pulse Fal'Cie spread. But if you read the last three analects, you get an idea of what is going on.

(SPOILER)
XI. Hypothesis of the Hunt
Obtain: Clear Mission 63: Crushed by Doubt
We've long held the goal of the fal'Cie's endless excavations to be the expansion of the world's inhabitable space-the creation of new lands with which to honor the gods. But I contend this to be false. Their methods lack the order one would expect if that were their purpose, and what's more, they gods they would honor have long since departed this world.
I propose a different explanation: the fal'Cie are hunting. Whether they seek a way to recall the gods or to journey to their side I cannot say, but I do believe the fal'Cie seek their lost deities. They search the earth, the skies, the waters, and even the deep places, seeking a gateway to the Divine.
-- On the Nature of Fal'Cie

XII. The Door of Souls
Obtain: Clear Mission 63: Crushed by Doubt
When our earthly vessels meet their end, the souls they housed must leave this world. Would the path of their migration not be the same one as our departed gods? Must they not pass through the same doorway the Divine employed to reach that place that lies beyond?
If this is the case, it stands to reason that, should a great many lives at once be cut short, a flood of souls would surge through the aforementioned portal. The Door would be thrown wide, and perhaps we might even glimpse the gleaming light of Divinity beyond.
-- On the Nature of Fal'Cie

XIII. Fabula Nova Crystallis
Obtain: Clear Mission 64: The Doomherald
Children of Hallowed Pulse scour earth, searching substance for the Door. Those of Fell Lindzei harvest souls, combing ether for the same. So have I seen.
The Door, once shut, was locked away, with despair its secret key; sacrifice, the one hope of seeing it unsealed.
When the twilight of the gods at last descends upon this world, what emerges from the unseeable expanse beyond that Door will be but music, and that devoid of words: the lamentations of the Goddess Etro, as She sobs Her song of grief.
-- Author unknown

So the Cocoon Fal'Cie believed that the only way they could bring back The Maker through sacrificing humans (XII) and therefore created Cocoon basically as a human breeding ground (that's why they basically created an utopia for humanity).

The Pulse Fal'Cie believed that they could find a way to find the Maker, hence their continuous exploration (XI).

The ambiguous question is whether the Pulse Fal'Cie wished to destroy Cocoon because they were in the same scheme as Barthandelus or whether they wished to get the raw materials of Cocoon (as it was created out of Pulse resources) so as to see whether the key to finding the Maker was in it.