How does C. S. Lewis define the "natural" and "supernatural" in his book Miracles?

From my notes on chapter 2 of Miracles:

Miracle: “An interference with nature by supernatural power.”

Naturalists: “Those who believe that nothing exists except nature.”

Supernaturalists: “Those who believe that, besides nature, there exists something else.”

Nature: Defined by naturalists as “everything there is.” More rigorously, the natural is “what springs up, or comes forth, or arrives, or goes on, of its own accord: the given, what is there already: the spontaneous, the unintended, the unsolicited.”

Supernature: “That which causes nature to exist.”

The naturalist's contention that "nothing exists besides nature" basically means that everything that has ever happened is explainable via natural laws (a viewpoint also known as "physical determinism"). The Supernaturalist believes that someone or something can intervene and subvert the natural course of events. From my notes on chapters 3-4:

In principle, everything that exists in Nature must be explainable as arising from elsewhere in Nature, according to the Naturalist. Anything that is unexplainable (not merely unexplained) from this standpoint would destroy the entire premise of Naturalism.

In chapter 3 or 4, Lewis mentions advances in quantum physics as evidence that others have put forward for the existence of the Supernatural. Many quantum events are to date unexplained, and, many Christian apologists argue, unexplainable. Lewis says that he doubts they are physically unexplainable (and therefore not a proof of Supernaturalism) but casually uses the word "Subnatural" to describe the hypothetical existence of unexplainable quantum events. He essentially says that such events would, strictly speaking, be "Supernatural" but he avoids using that word because it would be "a shock" to the reader.

Lewis says elsewhere in chapter 2 that multiple natures may exist. A fictional example of this would be his Chronicles of Narnia. If you recall, The Magician's Nephew establishes that there are many worlds (which are all separate universes), including our own world and that of Narnia. Each of these worlds was created separately and is governed by its own laws (Narnia in particular being a much more magical place than our world). At times, however, there may be divine intervention to bring the two worlds together, like when Aslan calls the Pevensie children from our world into the world of Narnia. In this example, our own world/nature is the "Supernature" to the world of Narnia, because things that existed in their own right in our world sprang into existence, from Narnia's perspective, seemingly from nothing. Similarly, when Aslan visited our world in The Silver Chair, there was no way for our laws of physics to explain his presence or his existence; in this case, his intrusion into our world was by "Supernatural power" (that is, power foreign to the "Nature" that we know, being from Narnia) and therefore "miraculous."

If I recall correctly, Lewis did not give a specifically Christian defense of miracles until later in the book. Therefore he does not mention God at this point. But God himself would be the ultimate Supernature, seeing as he exists entirely in his own right and for his own sake, is sustained by nothing, and sustains everything.

In Lewis' conception, heaven qualifies as a "Nature" and, when angels visit our world, heaven qualifies in such a context as "Supernature."

Lewis, if I remember right, acknowledges that his definitions are novel, but says that he defined his terms the way he did by necessity. There hadn't been a technical book on the existence of miracles prior to this one, and even the word "miracle" was defined by him in the book a little differently than it often is used in casual conversation. However, the ideas he expresses through his technical language are not novel, except perhaps for his idea of the "cardinal difficulty of the naturalist."