Wait wait wait....no hold on, those Eva explanations are kind of off a little from what I remember.
Adam is the first form of Life on the planet, or something along those lines. From Adam, Lilith is born, as are the other Angels who all for some reason go into hiding/hibernation etc.. who knows, maybe the Ice Age...did something I don't know.
Anyway, so Adam is the first Angel and Lilith is the 2nd Angel. That's why the attacks start with the 3rd Angel. The Geofront is basically the Garden of Eden, or at least metaphorically implied to be.
Anyway, so Lilith and Adam are seemingly extremely similar in almost every way. They seem to share all the same properties etc..
Anyway Lilith is the first (and only) Angel to have it's own offspring, that offspring were Humans (the lilim).
So anyway Second Impact happened because Humans found Adam in the antarctic, he was presumably frozen/sleeping/hibernating/in stasis whatever. They tried to turn Adam in an embrionic state, to control him, sample him, recreate him etc.. or whatever else they wanted to do. Adam awakens during this process and is pissed off, extends his AT field at full power across the planet which cancels out the much weaker AT fields of humans, hence a shitload of the world population dies. However the process is a success and Adam is transformed into an embrio.
However anybody born close to or on this date is for whatever reason deemed "innocent" or rather free of "original sin" (perhaps as they had nothing to do with the Adam thing, clean start for a new generation vibe or whatever), this is why only 14 year old children can pilot the EVAs. At least this is what is implied anyway.
Anyway Angels begin to awaken 1 by 1 and move towards the geofront as they wish to be reunited with Adam so that they may control what happens and remake the world as they see fit (presumably). It can logically be assumed that Angels believe Lilith to be Adam from a distance due to their similar nature, thus why Kaworu only realises when he is right in front of Lilith that it is not Adam. Although later on they retrieve Adam in embrionic form from Antarctica and he is also kept within the geofront.
EVA's are designed based on the genetic material from Adam, but Eva 01 is different as it is based on Lilith. However although they can copy Adam's genes, there is a missing piece of the puzzle to create a lifeform, a soul. In the initial activation experiments with prototypes the Soul and consciousness of the user is absorbed into the EVA, thus giving it substance.
This was unknown to Humans until they tried the experiments. Yui Ikari, was lost in the activation experiment of what would become EVA Unit 01. Thus her Soul and Consciousness exist inside the EVA. This is why only her son Shinji, or her clone Rei the "other her" can pilot Unit 01.
Asuka's mother was also involved in a similar experiment however it the effect was stopped before completion and her body was saved, but a piece of her was missing, her soul, or part of her mind or both were absorbed into what would become Eva Unit 02. This is why her mother goes crazy and why Asuka see's her mother in Eva 02.
It's unknown who's soul resides in Unit 00, but it has been speculated that the body of Ritsuko's mother was used for it's activation, mainly due to the fact she was dead and the fact that Unit 00 goes berserk and seems to be targeting Gendo specifically during it's activation experiments.
Also, Rei is suspected to be a combination of genes from Lilith and Yui Ikari. While Kaworu is made from Adam and who know's genes.
So anyway SEELE, with foreknowledge from the Dead Sea scrolls believe that humanity has reached it's evolutionary apex and that the only way to advance humanity is by creating one singular consciousness for all of humanity, in otherwords move on to a higher plane of existence as a singular being. They wish to manipulate events to achieve this, however they wish to control the world that is created afterwards, in other words direct the flow of human evolution to how they see fit, they wish to be the dominant consciousness.
Gendo also wishes to be in control of it as he believes that by merging all of humanity and the EVA's that he can be reunited with Yui who resides inside EVA Unit 01. However it is Shinji who is given final control by Rei when she instigates the 3rd impact by merging with Lilith.
So the 3rd impact would basically be every human on the planet having their AT fields shut down so that all of their souls and consciousness merge together and they lose their physical bodies, at the same time they will merge with Adam/Lilith the originator of this, who will control what happens next, however whoever merges with Adam/Lilith is the one who control's the flow.
That's about all I can remember, hope it makes sense.