I agree pretty much with your definition. What has happened over time is that many things that were supernatural have been explained and are now through to be perfectly natural. Earthquakes, lightning, meteors, etc were all thought to be magic until they were scientifically understood. Ghosts are considered supernatural, although people are studying them.
There are many examples of encounters with ghosts that are evidence of something being there, but what that is is not fully scientifically understood.
I used to watch a TV show called Ghost Hunters. I know many people have criticized it for being hokey. But on one episode they were in a building known to haunted. They had a camera crew and a sound man with them that were usually off screen. One incident however was recorded. The sound man, was carrying a large shoulder bag filled with stuff. The bag was hanging by his neck in front of him. All of a sudden, he was knocked unconscious. The video showed him walking and cuddly, the bad lifted up and smacked him right in the face. Now, assuming that was real and it really happened, is evidence of the supernatural.
I view spirituality as being different from being supernatural (I know that others disagree). I prefer to think of supernatural as phenomena that have some basis in reality that perhaps someday will become part of the physical world. I believe spirituality is metaphysical or beyond the physical world. Spirituality is something that can only be experienced by individuals in a completely non-physical way. In other words something that cane experienced but can never be reduced to science.