An intuition is just something you believe without being able to say why you believe it.

If you have reasons for your belief, then it's a conclusion, not an intuition. If you can't give reasons for it, then you can't argue that anyone else should also believe it, by definition.

"So what is intuition supposed to be? Rational intuitions are a spontaneous, rapid psychological assessment of truth and prompting to judgment about a priori propositions."

Here the author changes the subject from intuition to "rational intuition", but the latter just means a conclusion you come to quickly, and all the examples given are ones that you could readily supply an argument for. That's not the kind of epistemological starting point that people usually mean when they say they have an "intuition", and it's not what people object to when they object to using intuitions as "evidence". Intuitions aren't evidence of anything except the fact that you believe them. If you have reasons for your beliefs, give the reasons, that's it.