Perhaps you can't be thinking that your thinking nothing at the time you're not thinking it, but you can easily be not thinking and then someone comes along and says "what are you thinking about?" and you realise "nothing".
I am glad you realize this, because to me it seems central to the discussion and yet almost everyone just glosses over it. Especially when it comes to qualia.
Suppose you are standing in front of a Rothko. I love Mark Rothko, he is one of my favorite artists, and not many people know this but his
intended way of experiencing his work is to stand right in front of it, just a few feet away, so that your entire peripheral vision is filled with the painting. When you do this, something special happens which I can only describe as a feeling like you are
in the painting, or even that you
are the painting ( which is just a mass of color, so don't read too much into that ), and it is pretty cool.
So I like to use "standing in front of a Rothko" as the quintessential scenario for producing qualia in our minds. If you would ever experience "seeing red," it would be when you stand in front of a red Rothko.
Anyway -- you are standing in front of a red Rothko.
There are two possibilities for what you are "thinking," ( and I accept the sort of intuitive definition you gave ) -- either you are thinking about seeing red, or you are not. That "not" could include just about anything, for example say there is a cute girl next to you and you are utterly focused on her presence although you are pretending to look at the Rothko.
Now obviously when you are not thinking about seeing red you are unaware of the fact that you may be experiencing seeing red. And it isn't until someone asks you about it -- perhaps the girl, maybe she says "isn't this red exquisite?" -- that you
realize you were experiencing seeing red.
So clearly there is a distinction between those two mental states and how they relate to the quale of redness. The fundamental question of consciousness, to me at least, is how those mental states are different and what it implies.
And I think the answer is just right under our noses -- that you are
focused on the experience, actively
thinking about it, in the latter case but not the former. And since every case where we wonder about qualia involves thinking, and actually focusing on the experience of something like seeing red, it seems to me that explaining qualia is as simple as just thinking about an experience -- like thinking about seeing red.
What is "seeing red" without the "thinking about seeing red?" I don't know -- whenever I think about it, I am thinking about it, and whenever I think about doing it in the past, I am thinking about my memories of something I was not consciously aware of at the time, and I am not convinced those memories are reliably uncontaminated by thought. To elicit a memory and not have it contaminated by your current thoughts would entail somehow doing it without thinking about it, and I don't see how memory can be consciously accessed in such a way.
That is one of the reasons I don't agree with the "hard problem of consciousness" crowd. I think they are pondering how the experience of seeing red arises, and although they realize that they are thinking about it, they also think they can just extrapolate to some "thoughtless" type of experience and the experience would remain the same. I don't buy that, primarily because when I am not thinking about seeing red, I have no idea what it is like to see red. Maybe I am different than most people, but that is just the way my brain works -- when I am sitting here typing, the red portions of the screen are just another aspect of my visual field. It isn't until I look at them and think "red" that I am "experiencing seeing red."
I would love to hear your thoughts on this.