The principal use of the supernatural around here is in relation to religion, of course.
And the principal use in religion is to claim that beings not found in reality exist in reality ─ gods, angels, demons, souls &c ─ and to attribute magical powers and imaginary attributes to them, particularly to God.
Magic is the alteration of reality independently of the rules of reality ('laws of nature', 'principles of physics' &c). It fits your definition of 'supernatural' quite well.
The most striking thing, at least to my mind, about these claims is that they're made despite there never having been an authenticated example of the supernatural. It's never been found in nature, in reality, in the world external to the self. It's only found in stories, in imagination, in dreams, in abstract concepts with no real counterpart.
There isn't even a testable hypothesis as to how magic and the supernatural might exist, might work. (One of the great puzzles to me is why, if the churches believe in miracles, they don't have teams of experts financed by research grants, seeking to describe and define miracles and how they're done ─ the knowledge, were it real, would be hugely useful ─ like Harry Potter, you'll never have to change a tire, replace a broken window, walk anywhere, again.)
Of course, I'd be delighted to be shown to be wrong about the supernatural ─ but I suspect the first person to demonstrate real magic will get a Nobel prize or two.