Whereas it's true that we should be careful to not retro-apply characteristics to the Allagans by how the Garleans act or supposedly see the Allagans themselves, I was actually thinking of other evidence when I assumed those characteristics apply to the Allagans.
For example, we know with certainty that the Allagans fought Primal Bahamut, defeated him, and then set up a fancy set of devices to capture his essence and siphon his power. This is hardly the attitude of a group of people who have a healthy respect for the so-called gods of other races. We also know that the Allagans did not seem to worship any Gods themselves and sought to trespass on the plane of gods through research and accumulation of power. Xande even went so far as to try and harness the power of Darkness itself which, according to the lore, would be loosely tied to Zodiark (being the god on the dark side of the scale, opposite to Hydaelyn). It's a bit of a leap, but the Allagans had some lofty ambitions if they thought they could abuse the power of that particular god. From what we know of the Allagans, I think it's safe to assume that they saw the "Gods" as something to be eventually conquered by their Empire.
Also, Louisoix/Pheonix and Shiva were not the fist Gods to make us question the nature of Primals (well, they might have been the first to make the moronic NPC's question it, but not the players). Odin, technically, is the first, since he has been around the longest and we still don't really get how his summoning works with certainty. However, the ones that I find more interesting in regards to the difference between Eikons and Primals are the "Extreme" versions of Garuda, Titan, and Ifrit. Not many people follow that lore, but those three, supposedly average, Primals attained power far superior to their original incarnations not because of an increase in aether (though they still do require an obscene amount of it), but because of a ritual sacrifice. Similar to Odin, Phoenix, and Shiva, the Extreme versions of the original Primals required a living body to achieve their summoning at a level high above their original or Hard incarnations. This was the defining feature explaining their strength. Yet, no one has ever referred to them as Eikons, and I think that has a lot to do with the fact that these 3 Primals are bound to Eorzea. They aren't referred to as Eikons because those who used the term originally are no where to be found and the Garleans are from another country.
I agree that we certainly can't just take the Garlean's word for it, since we don't even know how much they know about the Allagans or the nature of Primals, but there seems to be more cultural and geographic evidence that explains the distinction between "Eikon" and "Primal" that there is to prove that they are somehow fundamentally different. Especially because some of those differences that people have pointed out have actually been shared with the so-called "Lesser Primals" when summoned in their stronger form. In fact, the only characteristic that we can say is unique among one confirmed Eikon, Odin, is that he is self sustaining (and even this may not be the case if the sword theory turns out to be true). Alexander and Bahamut do not share this trait, as they have confirmed summoners. So, so far, there's a lot more evidence pointing to it being a cultural/geographical/language distinction than there is to say that it is an inherent difference in nature.