How do you reason about reason? Would you rather I'd called those "system 2 and 1" or something? Or do you just find that none of the existing literature about human cognition matches your experience of yourself?
The heart pumps. The heart is not a pump. There is neural tissue all over the place, hormones by glands which affect decision making and so on.
You will not easily categorise kinds of thinking in any objectively measurable way.
Type One thinking may have 72 subtypes once we get good brain imaging technology. They may decide the original category was wrong And So On.
You cannot use those concepts, like type one thinking, except as heuristics.
They are not true like “the sun is mostly hydrogen” is true.