I'm no longer so sure that Eva-01 and Mark.06 are two of the Adams. This is the hypothesis I'm working on now:
The Confusing Meaning of "Angel"The word "Angel" is used with two different meanings in NTE.
Given how angels are numbered in order of discovery, it must be the case that humanity first found Kaworu (or some other physical entity hosting his soul, before he emerged on screen from the coffin in the shonen body we all know).
Next, they discovered Lilith and the Geofront and realized she too was a lifeform that could not be explained by current science, so the generic term "Angel" was coined to refer to such beings. At some point the connection was made with the Dead Sea Scrolls and her role as origin of all known life on Earth was established.
The next "mystery" lifeform found was Tunniel, which was accordignly named the Third Angel. It later became clear that the Angels discovered from the Third on are essentially different from the first two, and it is these ten entities to which "Angel" refers when Gendo explains the options handed to humanity by God.
So in a broad sense, "Angel" means "any superhuman, until know unexplained life form" (except the Adams, for some reason); and in a norrow sense, it means "The ten FoL-based beings that occupy a privileged place as God's children".
The role of Lilith, The (proper) Angels, and Third ImpactLilith herself is the beginning and end of all life on Earth (as per Misato, 1.0); as such, the Angels seek to merge with her and trigger a global L Containment field that will obliterate all FoK-based life. This is the "be exterminated by the Angels" option given to humanity by God. Misato refers to this hypothetical event as "Third Impact" near the end of 1.0, simply because it would happen after Second Impact, but it is in no way similar to the Third impact that comes next in Gendo's/Seele's plan for instrumentality.
In this sense, the behavior of the Angels is much more straightforward and easy to understand than it was in NGE. There is no "Adam" to return to, and they simply seek Lilith in order to exterminate humanity as a punishment for being FoK-based lifeforms, by God's command.
What Happened at Second ImpactThe Katsuragi expedition found 5 Adams and 6 spears. They were either lying there in the South Pole, or they were able to bring them out of negative space somehow (the latter is consistent with Gendo's exposition).
For whatever reason, they awakened the four lesser Adams using four of the spears. The Adams developed halos and turned into giants of light.
The four spears used were consumed, the awakened Adams "burned all the way up" (nobody stopped Second Impact, it extinguished itself once the Adams had been consummed). The resulting explosion blew one of the remaining spears to the Moon, the 'surviving' Adams all the way to the shores of Japan (chalk outline seen in 1.0? Although it has only two arms), and the remaining spear somewehre else but still on Earth.
The Earth-bound spear was recovered and pinned to Lilith for some reason. The other one was discovered on the Moon, and that's one of the reasons Seele built the Mark.06 there. The surviving Adams was kept dormant by either Nerv or Seele, for future use.
All production Evas (00, 01, 02, 03, 04?, 05, Mark.06, Mark.07, etc.) are likely cloned from genetic material harvested from the four Adams prior to Second Impact. In particular, Eva Unit 01 was cloned from the three-eyed Adams. When it awakened, it achieved pseudo-deification, meaning that it was the closest thing possible to the original awakeded Adams it was based on,
but not quite. In contrast, when Eva unit 13 awakens, Asuka mentions that it has gone "past pseudo-deification". That is, it has turned into an actual God, because unlike Unit 01, it is made from "the real thing".
On AwakeningIn NTE lore, "awakening" seems to equal "developing a Halo and becoming a potential impact trigger". The Adams (and the Evas cloned thereof) seem to need to undergo a specific process to awaken, while the Angels (proper, i.e. 3rd on) are in an awakened state "from the start". Gendo masterminds Near Third Impact at the end of 2.0 and Near Fourth Impact (unofficial name) at the end of Q, with the specific purpose of awakening Eva Unit 01 and Eva Unit 13 respectively.
Eva-01 awakens by the sheer will of Shinji. It later incorporates the core of the 10th Angel (delivered by means of Rei Ayanami, later revealed to have been "built specifically for that purpose") and turns into a radiant giant, but the awakening happened earlier.
Eva Mark.06 Seems to be awakened from the get-go; perhaps this is by virtue of it having an (artificial) Fruit of Life and/or being the "true Evangelion"/"Actual God, not a fake one".
Not sure what to make of it, but there is an intravenous bag labeled "BTB" ("blood type blue"?) at the construction site.
Eva-13 awakens by consumming the 12th Angel. Although arguably it too awakens by Shinij's will, and only turns into "radiant giant" when the angel is consummed: When Kaworu tells Shinji that Eva-01 awakened and became the trigger of 3I, we are shown the cut where its eyeballs bulge red out of their sockets and the red L Containment Field rings first appear overhead. The red eye-bulging in Eva-13 happens when Shinji cuts Kaworu off the controls and starts climbing Lilith's carcass (albeit without rings / doors of Guf overhead).
Eva-02 awakens by injecting "angel blood" and/or its pilot (again, "built specifically for that purpose") unleashing the 9th angel within the entry plug.
It seems like obtaining the FoL could be tied to awakening, but not quite (see Eva-01); so it's not immediately clear how the four Adams were awakened during Second Impact. They did each get an Adams-type halo for sure.
Eva-13 has a double (tangent) halo. Perhaps because it has two pilots/souls manning it.
On the HalosBack on topic of the halos, the Wunder has two "Adams type" halos. Likely, one is from Eva-01, and the other from the absorbed 10th Angel, which manifests as Adams type by virtue of being subdued? Or maybe (out-of-universe) one halo of each type didn't look good visually
In contrast, the 3 Nerv ships have a single halo each, purplish, of "Angel type" if I7m not ,mistaken. Can't remember what the Adams Vessels halos look like in 3.0 and 3.0+1.0 (I think Angel-type), but after Eva-08 "overlaps" them, the four halos appear to be of Adams type. Not sure what to make of this...
Addendum (2022/Feb/07)The Mark.06 halo totally looks angel-like (sharp edge, rainbow-ish colored), not Adams-like (fuzzy edge, fully white). I wonder what that means...