A lot isn't clear at this point - Gendo wants to initiate the third impact to, presumably start the human instrumentality project for everyone but him, so he can be God.
From what I gather, having two Longinus Lances instead of a Cassius and a Longinus was the crux of Gendo's plan. The Longinus Lance is designed to eradicate any threat posed by the Angels as a failsafe, and the Cassius does the same for the Lillim (Humans). So, I think, what Kaworu was trying to do was to end the conflict between the Angels and Lillim once and for all, but instead he realised that what Gendo wanted was to only destroy the Angels and enact the Human Instrumentality Project - and combine all consciousness into one, except his own.
I * think * the fourth impact was stopped was because Lilith chose to close the gates of Guff, because she is still possessed of the consciousness of Shinji's mother, but in doing so, this erased Shinji's mother from the Lilith entity.
I think Rei's Eva is called the Adam vessel is because Adam is the father of the Angels, while Lillith is the mother of the Lillim, so even though it's an Evangelion, it's essentially an Angel because it's furthering the cause of what WILLE sees as the Angels (and Gendos) cause - destruction of humanity.
I did quite like the third installment, despite how much it deviated from the original source material - but I wanted more of Mari and Misoto, and I wanted everything to be desmystified a little - Im also disappointed in Shinji's final action - which, unless it was instinctual and to protect his mother ,really just exemplifies what a tantrum throwing navel-gazing little turd he can be.