Why is Lillith misidentified as Adam? Basically, I'm rewatching for the first time and I just finished episode 15 where Kaji shows Misato Lillith (in terminal dogma right?). But he calls her(?) Adam. And I'm pretty sure Kaworu misidentifies her as well. Do all the other angels think Lillith is Adam as well? From my understanding, after Asuka is introduced, Kaji brought Gendo the embryo of Adam and at some point puts it in his hand? Not sure when that happens because it isn't shown iirc. Then Rei absorbs Gendo's Adam hand into her abdomen, flies up into Lillith's breast, and initiates Third Impact?  So Adam is literally in Gendo's hand, and everyone just thinks Lillith is Adam? Why?

This show is really confusing, and I'm really OCD about fully understanding a show before moving on to another. So any help, clarifications, and information is highly appreciated. Thanks!