I'd like to try to cram all my Ilberd → Shinryu replies into one post (and perhaps quote it from here if needed, lol).
Saying up front that we can (and should always) debate ways we might
disagree with what the game wants us to believe, what the game wants us to believe is quite clear, in my humble opinion. Most of it is spoon-fed via exposition, unfortunately ... but it's there.

Originally Posted by
Ilberd
Did you hear their cries as victory was snatched away from them? Even with their dying breaths they cursed the Empire! Never has their desire for vengeance been so raw, so true! A god has no need of faith when summoned by so pure a purpose!
Unlike most primals, the summoning of "Shinryu" does not include a focused ritual. There is no group prayer to a god (or it probably would have been Rhalgr). "The Griffin" led the Ala Mhigan Resistance to what they believed was an assured victory, and then set them up to be slaughtered in the most demoralizing way possible. Their assured victory was
taken from them when they saw the Empire's reinforcements - you can see their sorrow as the tone of the scene shifts when the magitek arrive. The only sentiments present are soul-crushing disappointment and the pure desire that the Empire will eventually suffer.
That is what Ilberd harnessed.

Originally Posted by
Ilberd
Oh, yes, I know their limitations─which is why I will call upon a deity more terrible than the very black wyrm of the Calamity itself!
This is pretty much the extent of Ilberd's intent: "Worse than the Calamity, please."
Think back to Louisoix for a moment. If watching the world become engulfed in flame and dwelling on a desire to see it reborn results in the image of a phoenix (and what scholar would
not think of the phoenix when dwelling on rebirth from ashes), what does a man who witnessed the Calamity think of when he imagines,
worse, please? (Especially when he's holding the eyes of Bahamut's brood-brother, which long seethed with nothing but a desire for pure vengeance and destruction.)
Just in case you don't arrive at this conclusion on your own, Alphinaud and Krile make sure to press it.

Originally Posted by
Krile
It was like watching a nightmare unfold before our very eyes. Ilberd's primal manifested in the form of a colossal dragon─a being of pure violence. It burst forth from the cocoon with such terrible force...

Originally Posted by
Alphinaud
That such a horror should spring from the eyes of Nidhogg comes as no surprise. Nor do I wonder at its form. Ilberd all but announced it in the moments prior to his death. Plainly, it was his dying wish to visit a second Calamity upon the Empire.
They're trying really hard to get you to arrive at the idea that Ilberd simply reproduced what he saw in the Calamity, and the Eyes were the perfect tool with which to do so. (For the record, yeah, it does kind of bother me that they rapidly mashed the 8-Man Raid themes into the main scenario to entangle the stakes and get us all hyped up only to require Alphinaud to explain why those themes' origins should be
so obvious.)
Merlwyb and Kan-E-Senna also go out of their way to let you know that the downing of the imperial airship and the sights they witnessed from afar just made them think
Bahamut over and over again.
So while we should of course contemplate whether or not these assumptions are accurate, I think what the game wants us to think was nicely summarized.
Also, indeed, Shinryu is (as far as everyone present is concerned) a convenient temporary label. The fact that Omega vs. Shinryu is just
where it's at if you want a Final Fantasy themed amusement park (which Yoshida-san absolutely does) is a "happy coincidence", lol.

Originally Posted by
Kan-E-Senna
While most among us could think only of Bahamut when looking upon the primal's form, the Domans were heard to whisper the name “Shinryu.” It would appear that the being resembles a creature of Far Eastern legend, and we have found it convenient to refer to it as such.