The first one, "Ignorance", is said of an individual misinterpreting reality.
You got it correctly, that is not the right definition of Ignorance for the purposes of D.O. When we discuss D.O. we are speaking outside of a context of an individuum. Instead, in the impersonal context where D.O. takes place, Ignorance is a primordial lack of representational differentiation. Nothing is represented, nothing is differentiated, nothing is known. You can imagine it as a primordial soup of randomly fluctuating energy currents, forming some shapes bouncing against each other and falling apart.
The second one, "Formations", is where the individual performs actions [...] How exactly does consciousness arise out of "unconsciousness"?
Once again, there is no individual yet, no one to perform actions. Instead, as those random energy currents fluctuate, some of the shapes they form end up getting "dynamically persistent" - meaning they are not strictly static and yet they tend to get cyclically reproduced due to being parts of the causal loops. This does not happen everywhere, only in the pockets of low entropy that can fuel the dynamically persistent shapes.
These shapes or Formations bear imprints of the forces that shaped them, carrying information about them, thus being the first very primitive instances of Representation. Over time, after many, many, many cycles, many iterations of the causal loops that shape them, these dynamically persistent formations get combined into clusters of dynamically persistent formations of the higher order, giving rise to more complex Representation.
Eventually it gets to the point where these Meta-Formations grow capable of a kind of mechanical response to certain patterns in the stimuli of the energetic fluctuations they interact with. These mechanical responses are the ontological substratum of Recognition of Signs, the basis of Semiosis. Over time these stimuli-response pairs get combined into chains, that get more and more complex, until there it becomes what we can call a very primitive kind of Consciousness, an advanced kind of pattern recognition.
All this is to say that, in its basic essence, Consciousness is nothing but an emergent property of Formations being carriers of Representation.
So far our discussion was taking place in some abstract cosmological space, before the primordial soup of energies has developed anything we can call life, let alone sentient life. In our segment of the Universe that is clearly not the case. Instead, on this planet we have a booming ecosystem of various causal loops forming dynamically persistent entities known as biological processes and life forms. As I pointed out above, all of these fall in the category of what Buddhism calls Formations. And, as I explained, these Formations inherently carry Representations of energies that shape them. I hope this makes it clear why these Formations should be considered inherently conscious in the very abstract, very rudimentary sense of the word.
Now, as you can imagine, some of these formations evolve to be capable of more complex and more cohesive Representation than others. At some point this Representation enables the level of Pattern Recognition or Semiosis that can itself act as a positive factor of causal loops -- that is to say, a positive factor of natural selection. In other words, the Formations capable of Pattern Recognition receive evolutionary advantage. They no longer randomly bounce around, they tend to gravitate towards desirable causal situations, by taking advantage of Pattern Recognition. They avoid negative causal influences and seek positive causal influences. In other words, it's almost like they (the Formations) act with greed towards the desirable and avoidance of the undesirable. Although, strictly speaking, Formations and their Actions are not subjects and verbs - remember, Formations themselves are dynamically persistent causal loops (and meta-loops of many levels). So all of it is just activity upon activity, causal loops, also known as the karmic tendencies.
To summarize what we covered so far, the primordial soup characterized by Ignorance gives rise to Formations, the Formations give rise to representation, desire, aversion, and consequently, Formations end up gravitating towards what to them are desirable states and avoiding undesirable states. The causal consequences (karma!) born from this pursuit and avoidance feed back to this overall process, creating evolutionary pressure towards more and more capable Formations with stronger capacity for Representation and Pattern Recognition.
In other words, the karma created by the afflictions of ignorance, craving, pursuit, and avoidance, causes the seed of Representation to grow into more and more capable Consciousness. This consciousness is said to dwell in the impressions produced by karma, which are then deposited within it as representations of patterns or "objects".
The above may seem like it describes the evolution of Consciousness during the early part of Earth History. However, it is important to understand that sentient beings, including humans, are themselves causal loops of dynamically persistent entities, and as such every iteration of the loop repeats the key steps of the evolution, thereby reproducing itself. In other words, every sentient being goes through the same process where it starts from Ignorance and develops capacity for Representation and Pattern Recognition by growing certain Formations supported by its body and brain.
Consciousness, thus originating, has the role of delineating individual objects, the nama-rupa. The rest of D.O. describes how this ongoing process leads to the representation of "outer" vs "inner" and the eventual delineation of Self.
I think this should be enough. I hope this will make sense to someone.
