I agree. I think intuition is a word that used far too seldom, these days. Intuition is where you’ve grasped the solution before being able to prove, more methodically and painstakingly, that it
is the solution. I imagine that, when a scientist has an intuition, he follows it as a detective follows a lead; sometimes he’s proven right, and sometimes he comes to realize that his intuition (his “hunch”, even) was mistaken (or perhaps it contained a grain of truth to it, but only in part).
I personally don’t see intuitions of a more religious nature as being provable in this way (for me it is more of a private, indeed intimate affair), which doesn’t mean those intuitions can’t one day be proven right (to use a homely but quite congenial example, I quote a stanza from Kermit the Frog’s song the Rainbow Connection

).
Why are there so many songs about rainbows
And what’s on the other side?
Rainbows are visions, but only illusions,
And rainbows have nothing to hide.
So we’ve been told and some choose to believe it
I know they’re wrong, wait and see.
Someday we’ll find it, the rainbow connection,
The lovers, the dreamers and me.