I agree that those are useful words to distinguish. I also think it's worthwhile to have a thing called "rationality" which maybe means "doing the best possible thing with your brain given your goals," which has to include both of what we're calling reason and intuition [1].
But I wanted to say some words here to flesh out the subject in a little more detail.
"Reason" and "Intuition" may be separate, incomparable types.
I think of Reason sort of like a technical spec -- a description of how a system should work, which leaves the implementation details as an exercise for the reader. I think most people use the word Intuition, meanwhile, as a catch-all to mean "anything that's not consciously executed logical reasoning (and probably not sensory processing either)."
Underlying all these are the implementation details I mentioned I above--the actual "algorithms" our brain is executing. How you actually implement "reason" can make it better or worse, faster or slower, and I've found that people can make significant gains by changing the details of what they are doing with their minds.
More broadly, I think people are capable of having significantly more access to their "unconscious" than most people realize (hence the scare quotes around the word ;)), and further that those nonlogical processes can also be different, and yield better or worse, faster or slower results.
(To justify "significantly more access" a little: consider that 200,000 years ago, no one knew how to do formal logic, but all the required machinery was there between our ears. We did stuff back then that was a primitive type of reasoning just because we had no basis or training in doing better. Today we have decades of training and a culture that supports such things. If you've been educated at all, you've had years of training in reasoning. How much training have you had in introspecting on what we're calling here "unconscious" processing? Probably none or not much! I claim that training in this area can allow a person to have much richer and higher quality "intuitions.")
Anyway, many people sense that the processes they are executing aren't reliable, and some of those come to believe they must rely on what they think of as reason, to the exclusion of their other faculties like emotions. That makes sense, but I think the stronger move is to understand, in detail, what your mind is actually doing, learn what those processes are good for, and where they are weak, and finally change and improve the processes where appropriate.
There's a lot of interesting work going on in this field, and we're just getting started!
[1] I'm the managing director of the Center for Applied Rationality, a nonprofit in the bay area, so I spend an unusual amount of time thinking about this stuff.