The Ascions are an interesting bunch.
*Warning, MASSIVE FF Tactics spoilers here.
[SPOILER] In terms of their own "lore" again, SE is taking ques from FF Tactics. Each Ascian holds the name of a Lucavi from that game. In Tactics, the Lucavi were demons (each represented by a zodiac sign) that sought to revive their leader, Ultima, the High Seraph. The revival of Ultima required a very large blood sacrifice. So there are some parallels between the Ascians in FFXIV and the lucavi in tactics.
Also, the Lucavi have a false history in Tactics. The public record of the Zodiac stones counts them as "holy relics" that once belonged to the Zodiac Brave knights whom used their power to defeat a demon (presumably, Ultima). However, during coarse of the game, Ramza discovers a secret scripture or gospel of one of the followers of the Zodiac Braves that tells their true history, explaining the corruption of the crystals and the Demons they house.
In FFXII, this history is explained in a little more detail. It is stated that each of the Lucavi were created by the gods as a means of controlling or ruling over one or more specific parts of existence. For example: Mateus, the Corrupt, was born in opposition to Lahabrea, and was referred to as the Abyssal Celebrant and Scion of light. However, each of these godly creations were eventually corrupted either by their own job, like Mateus, or turned against the gods by Ultima (originally tasked with guiding souls to heaven and aiding in their resurrection) who led them in rebellion.
The only one who was not involved was Zodiark, who, though not explicitly stated as such, we can assume was born in opposition to Elidibus, as Elidibus was the one who originally summoned him in FF Tactics (though, under the name Zodiac, instead).
It is also noteworthy that while Elidibus is in FF Tactics, he stands apart from the remaining Lucavi. He is not part of the main story, and seems to have no relation to or interest in reviving Ultima. Likewise, in FFXIV, Elidibus stands apart from the remaining Ascians. Though, he seems to be far more attuned to Zodiark than any of them. If anything, it is implied that Elidibus is following the "True" path of the Ascians, doing only what is necessary. The remaining Ascians, similar to their Demon counterparts the Lucavi, have become corrupted by their own power and "job." Their goal is the same as Elidibus's, but they seem to have become lost in the many millennium they've been doing this. They take too much pleasure in the work and have fallen from their true path in favour of gratuitously causing choas for the sake of it. I have a feeling that they are misinterpreting Zodiark's will/needs. I also have a feeling that the "revival" of Zodiark requires a more specific ceremony that the other Ascian's have been ignoring or forgetting, and that only Elidibus is truly in the know. [/SPOILER]