Use "avatar" if you want, but the whole Rei-Lilith relationship never felt very avatar-like to me. There are enough variations on avatars in Hinduism, but the Rei-Lilith relationship isn't in them.
Very well, I'm not going to insist on its usage, I just mean that it provides a ready example of a deity having multiple mortal personas who are still definitely the very deity.
While it is possible, even likely, that Lilith was deceived in some fashion to yank her soul out of her body, Lilith was never really dead. A dead body doesn't bleed LCL or regenerate lost legs. Adam, likewise, blew up but even with her soul in Kaworu, her physical body was still regenerating as an embryo. Saying that Lilith "died" here is a stretch.
Additionally, we don't even know what the Dead Sea Scroll "prophecies" say in the first place or how they're interpreted. The DSS are a complete black box since none of the show's creators have ever directly revealed what they contain (as opposed to what Seele interprets or outright makes up). Assuming that they prophesize the exact fate of every single Angel is going a lot further than predicting aspects of their appearance, role and abilities based on predetermined conditions. Had the Dead Sea Scrolls prophesized the deaths of all the Angels, it would make Nerv's task a heck of a lot simpler.
The listing of the Angels is clearly not in the order of their appearance or creation- otherwise Lilin would be the Third Angel. Even if you suppose that Sachiel through Armisael were created by Adam after Lilith's creation but before his sealing by the Lance, that can't apply to Tabris.
Instead, the numbering is very clearly in the order of the Angels' deaths. You can replace "death" by "removal-of-the-soul-from-the-body" if you wish to be technical about the actual operative factor. And this order is predetermined- Seele refers to the Thirteenth Angel (Bardiel) ahead of time, knowing that it will be different from those before. In fact, this is the only thing that ever indicates that they have any knowledge of the individual Angels' natures ahead of time.
How this works in regards to a potential Angelic victory scenario, I'm not sure. It's possible that the Angels' deaths are
actually predetermined and they were never capable of victory, and Gendou's & Seele's measures are merely to prevent them from damaging Lilith/Shogouki, whom they'll need for their Instrumentality scenarios. Alternatively, the DSS permit multiple paths, and the order we see is merely that of the branch that Gendou & Seele are trying to keep things on.
To clarify, I'm saying that it's unclear if the "scenario" is more of an explicit branch of the DSS prophecy Seele is trying to stay on (i.e. the DSS says something like "if Lilin is to be the last Angel, the other Angels must be killed in this order:"), or if it's their own plan built around a single prophecy (i.e. the DSS says effectively "Here's a list of the Angels that will be killed in this order:" with everything else implicitly open.)
(The explanation given in the CI, of the DSS as a manual regarding FAR technology rather than actual prophetic knowledge, is grossly inconsistent with what's in the actual show, and must discarded.)
This is an interesting idea, but why should the Rei-Lilith being preferentially take Rei's form during instrumentality? Lilith appears as Rei before Shinji in Episode 01 before Shinji or the audience know her as Rei. After collecting the souls of dead Nerv employees (who presumably don't all know her personally) her various manifestations take on the form of Rei. Lilith's very identity has been reshaped here.
Because Rei is the form that bears the genetic link to Shinji, the very reason she's probably appearing to him to begin with.
And what other form, I'm going to ask, should Lilith be taking to deal with Lilin than that of the Lilin existence she herself was just in?
As for whether her fundamental identity has been reshaped, that's a question that I think is beyond inference from the text, and also depends on what you mean by it.
Rei I gets 50% of lilith's soul while 50% stays in lilith's body.
There's no reason to think that souls are homogenous when divisible, or that the Lilith-body has retained 50% of the soul- Rei can certainly do a lot more than the original body can, and I'd expect otherwise if they were evenly split like that. Moreover, if there were a substantial element of the soul left in the Lilith body, then I'd expect that to come out when further Rei bodies were drawn out- instead Ritsuko explains that only one Rei bore a soul.
Rather, the division of the soul is functional
- Lilith retains only a barest fragment if the lowest, vegetative level if any. The element in Zerogouki is presumably some violent, animalistic element- the id or the like. But the element in Rei II is the crowning element, the rational soul, that capable of intelligent thought, speech, reasoning; that which contains Lilith's being-as-a-human (rather than as a plant or mere animal). This is obviously the most important part of Lilith's soul, and this is the one that Rei has. So she clearly is spiritually Lilith, Lilith-the-human-being (if not Lilith-the-giant-space-creature) on the fundamental level.
Why she lost her memory while Kaworu retains his is hard to say and rather beside the point. Plausibly memory is mostly physically stored in the original body's brain, and doesn't transfer with the soul very well (this also explains why Rei III seems a bit out of the loop). Indeed, it's not clear that Kaworu actually has
retained his memory, rather than being just told by Seele who he is- which Gendou neglected to do with Rei.
(Division of the psyche by function seems to be the norm in Eva. The CI states that the Nigouki contained the maternal element of Kyouko's soul. Naoko divides her soul by three interpersonal roles she holds. Leliel talks about an acting function of the soul and a critiquing function.)