I just had a mad thought. What if the Kingsglaive need to go through some kind of magical test to prove themselves worthy of sharing the Caelum line's powers and being part of the unit... That thing could be part of the challenge, hens it being part of their insignia and Prompty being in Kignsglaive attire. It could also be an entirely optional part of becoming a Kingsglaive member.
Don't really think its Ifrit. Way too skinny. Probably something quite innocuous in the final game like "Spirit of the Glaive" or something like that.
Something also to consider is that even Nyx has connections to the weird Daemon thing we see on the pin in some way. Note his attire in Kingsglaive. He has one horn that sticks out around his back/neck area on one side that resembles the same type of horn, of course much smaller.
Is this some shared connection with the pin? Why does only Nyx have it? It seems interesting to me that Nyx is known as this great warrior, and that perhaps that might explain his unique costume, but not its purpose.
As for my opinion on the daemon creature, I think it's just a daemon. I feel like all of the major summons in this game are going to be massive as they are guardians of the planet. It was also heavily implied that Ifrit lives in the Volcano near Solheim and would make a lot more sense than him randomly showing up in Insomnia. The astrals don't care about human affairs, they only care to protect the planet and have a much higher level of thinking than humans.
The only conclusion I can come up with as to why Ifrit would be at Insomnia is due to being forced to go there. The only way that could be true is if someone else convinced Ifrit to fight for their cause, but there doesn't seem to be any real possible way this can happen.
Keep in mind that the Astrals can speak to Noctis through his thoughts. Even back in Versus XIII days we saw the trailer with Noctis and Stella (RIP) talking and after she left we saw a mysterious unknown voice speaking to Noctis which couldn't be explained. It's likely that this is the same phenomenon that takes place when an Astral speaks to him. Whether that being was some sort of God or an Astral we can't say for sure, but one thing we do know is that the Astrals can speak to Noctis in his head.
Finally to touch back on the size thing, I have seen the counter argument used with Carbuncle being a small summon, but I believe that to be more due to the fact that Carbuncle is a summon that not only is designed to be much like how it's presented in other games (rather small and cute), but also that it will be able to be summoned in any location regardless of whether or not you're inside of an enclosed area like some building or a dungeon. One of the things Tabata mentioned was that when Astrals are summoned while inside of a dungeon they would perform a different animation like Ramuh throwing his staff instead of what they would normally do. Carbuncle is an Astral that could be summoned regardless of location and likely only do actions like buff the party or heal them up now and again, much like in the Platinum Demo.
Just to reiterate, I believe that only in special cases like Carbuncle would it make any sense for any of the Astrals to not be massive creatures. Think back to Episode Duscae two years ago (can you believe it's been that long) when after summoning Ramuh the other party members say things like "now I know why the tales are told".
These Astrals have likely been mostly myth and legend for the longest of time as they seem to all be sleeping hidden away until the events of FFXV take place. Leviathan is said to live in the waters around Altissia out of site. The fact that your party members would comment about the legends of these Astrals would imply that man has likely not seen these things in actions in hundreds maybe even thousands of years. However, they know that these creatures exist and that they are guardians of the planet, so each of the nations worship them in a way that pays respect to where each of the Astrals is said to protect, which explains why each nation has a sigil of a specific creature on it and has been the basis of many speculation for years now.
If these creatures are guardians of the planet, they need to be quite big, Carbuncle being an exception as it seems to be tied with dreams, and therefore serves a different purpose. I fully expect the major players like Ifrit, Shiva, Ramuh, Titan, Leviathan to be massive creatures (some of these already confirmed as we know).
So, that's my take on all of this.