Cocoon was created by evil fal'Cie (not Lindzei himself, to my knowledge) in an attempt to bring back the Maker, supposedly. I'm unsure if Lindzei definitely had something to do with it, or if it's just a case of Gran Pulse folklore of "Lindzie created Cocoon" which could be on par with people on Earth saying "the Devil created _____".

It's possible that they are actually not Lindzei's fal'Cie but rather fal'Cie that are branded to be "Lindzei's children" just because they are of a different viewpoint to those on Gran Pulse (sort of like one religion calling another religion "the devil's children" despite believing in the same God).

Alternatively, it's possible that they are trying to bring back Lindzei rather than the Maker, I guess. I don't actually have my guide or the game on hand at the moment, I'm just spouting theories.