Wait a minute...is this a plot-hole (Austerities of Wind and Main Scenario-related)?

[Kaiser-Ace]

I think you're presuming that there's a difference between aether and crystals, which is incorrect. When Louisoix tried to summon the Twelve, there were no crystals, and subsequently when he became a primal there were still no crystals. Personal and external aether shouldn't be the defining points of the two summoning methods. I disagree that there should be a differentiation at all. We provide less aether and therefore a smaller, more controllable size for the Primal, which we call the Egi. It has the Primal's own energy in it, instead of worship, but if we interpret those two to be similar, then the only differentiation between these two methods is the size of the summoned Primal. Further, during the summoning animation, the Summoner crystallizes his aether into three crystals that float around him. This shows us that we do still use a "crystal conduit" as you put it. These crystals, and the crystals used during a regular primal summoning, are what gives the Primal its mass. Therefore, I believe that we can conclude that aether should not be a defining point for differentiation, if there should be any differentiation at all.

The Crystal is a Conduit FOR the Aether, Louisoux didn't use either the allagain or ascian methods, he used the symbols of the twelve as the focusing conduites for mass prayer. How ever him turning into Phoenix was not the original idea it was a last resort back up plan that he literally came up with on the spot.

And again we still have no idea what primals really are, I'm just using what I've found out so far on the properties of summoning and the rules and laws of magic to come up with a theory. I've made the distinguishment between the Ascian Method which summons the Full God (Or desired being in Gilgamesh's case) and the Allagain method which summons a lesser incarnation of that God due to their fundamentally different properties, Ultimately a summoning is a summoning be it a Beast Tribe Primal, a Summoner's Egi, or Arcanist's Carbuncle. It's how it's executed that's different.

Louisoux and Iceheart form the biggest unidentified variables at this point. I'll refer to it here out as the "Incarnate Method" where one literally becomes the primal it's self. A small difference is Iceheart lives afterwards while Louisoux does not. After becoming Phoenix his physical form dissipated into raw aetheric energy and he and the primal he created became one and the same. Iceheart how ever still lived after Shiva's defeat and The Primal Shiva and Lady Iceheart do have several different personality traits and even speech patterns.

So there seem to be three possible Summoning Methods that allow a primal type being to be called forth. But until we beat the answers out of the Ascians on what primals are to being with all I can do is use what evidence I have. That being Egis are literal fractions of true primals, a single summoner does not have the Aetheric ability to summon a primal on their own and live, Crystals seem to be required as a conduit to focus more than one beings prayer into usable aether in order to perform a full summoning, and the Incarnate summonings of Shiva and Phoenix used the summoner's physical body as a catalyst for the primal with an as yet unknown variable causing Iceheart to live but Louisoux to die.

The definition of conduit does not imply Crystals are anything but what they are. We've used them as Conduite BECAUSE they are crystallized Aether. Thus making them a perfect conduit for focusing the unharnessed aethir of various sources.

Edit: (Additional Data)
A possible variable in the Incarnate Method could be that Iceheart has the Echo and can thus keep her soul separate from the Soul of Shiva when performing the incarnate Summon. Louisoux did not have the echo but with that power it's self being filled with unknowns for the time being that idea is inconclusive until we have a better understanding of how the echo works.