There is no precise language for this though, and if I clarified everything the comment would be too long so nobody would read it. By spreading out people asking for clarification like this we actually get to talk about it in a natural way.
> If, as you say, you are aware of something, it is by definition now in your conscious mind
Yeah, but that is a trivial statement, we all know that "thoughts popping into our head" means that they came from somewhere, I'm not talking about that effect.
To me the difference is like between a function call and a process. The dream is some other part of the brain that wants to know something that you are good at answering, so it asks you that question by posing it in a language you understand, ie the "dream", and then you answer by reacting to the dream, it isn't waking you up it is just making a quick invocation. This is different than what you are talking about, unless I misunderstood you completely.
When you are awake it is more that you are querying the other systems and they put thoughts in your mind as responses. Those responses will be calculated before you are aware of them, creating the "you made the choice before you think you did" effect, but you are still querying them. While when you dream those systems query you and you answer back with your reaction to the dream, meaning you have basically no control at all, and the shape and feel of the dream, when to dream, what to dream etc is decided entirely by that other part.
Or in other words, "dreaming" would be the feeling of being the subsystem others are querying, while "awake" is the feeling of being the main system that is querying others. And just like you ask your subsystems a lot of questions for all sort of reasons, I don't see why our other systems would have just one reason to ask us anything, they probably ask all sort of things for all sort of reasons, as long as querying you for information is useful to them they will do it.