Intuition is another of those terms on which it is almost impossible to
reach a shared understanding, so off we go again into the wild blue

I prefer myself Susanne Langer's explanation.

Consider the schoolroom syllogism:

All men are mortal.
Socrates is a man.
Therefore Socirates is mortal.

Suppose someone acknowledged that indeed All men are mortal and that
Socrates is a man, but could not see that Therefore Socirates is mortal.
That represents a failure of intuition.

There is nothing mystical or poetic about intuition; it is just a very
elementary stage in cognition.