Great question, Brad! In fact, I ask a similar question in debates with atheists whenever they claim that ID cannot be science because it posits a supernatural cause. In these cases I make my question rather specific. I ask them if they would consider the physical effects produced by intelligent agents (smart phones, for example) to be the result of natural causes or supernatural causes. If their answer is “yes,” then I point out that we have a good amount of empirical evidence for the supernatural. If they answer “no,” I point out that they have conceded that physical effects produced by intelligent agents are not supernatural. But if they were to say “both,” as you have, I would say that they are committing a logical fallacy.
I agree with you that when we believe in God, there is no such thing as a purely natural process. Nonetheless, I think we could agree that God has given us a universe that we can discover and learn about because he has given us natural processes that are constrained by natural law and fundamental forces. These purely natural processes can explain much about the natural world. It is precisely when someone, be they atheist or Christian, makes the claim that these processes alone constitute a causally adequate explanation for the novel body plans of living organisms, that they have declared a clear position.
Intelligent agents routinely constrain matter within the laws of nature and deliberately steer events toward intended outcomes. Thus, to answer your question directly, a supernatural process is a process wherein deliberate choices are made at successive decision nodes in order to steer events toward a distant goal. But if you prefer the word “event” to the word “process,” that’s fine. It works just as well. When I see a random arrangement of pebbles on the ground, I think nothing of it. But if I should see on the ground an arrangement of pebbles that spell out, “Brad Kramer is a Genius!”, I would be justified in my belief that this arrangement was the intentional result of an intelligent agent; that it was not the result of purely natural processes. The former arrangement of matter was the result of natural processes (at least there would be no reason to believe otherwise); the latter, the result of a supernatural process.
The natural processes that allow us to learn and understand our universe also happen to be necessary to create a universe capable of supporting advanced civilizations. They are not, however, capable, in and of themselves, of producing physical effects such as smart phones and skyscrapers. Smart phones and skyscrapers can only come about through a process involving intelligent agency. Similarly, apart from a commitment to materialistic explamations, there is no reason to believe that a life permitting universe could ever produce living organisms, given the advanced information and engineering protocols necessary to produce and maintain even the simplest known living organism.
Many contributors to this thread have declared that life is like rain or any other natural process,and that just as we do not need to invoke God every time it rains, so it is with living organisms. Now I am not saying that this is your view, as I do not know your exact view here, but such a view promotes life as a decidedly natural - as opposed to supernatural - process. I would say that because of the information and engineering of life, life requires a supernatural - as opposed to natural - process. It cannot be both. The very definition of the word “supernatural” makes this clear.