Fabulous thread, OP! I agree with a PP's comment about telepathy. Although I do have a healthy dose of skepticism about what I believe to be the more far-fetched stories (although there is wiggle-room!), I believe 100% in telepathy.
My stories are pretty tame, but....
When my grandpa was a young man in pre-war Central Europe, he had gone on a tram to visit his father in hospital. On his way there, he saw his father on the tram passing in the opposite direction, so grandpa got off the tram at the next stop and went back home, believing him to have been discharged. Back home, his father had indeed been discharged, but hadn't actually travelled on a tram. But grandpa had been convinced enough he'd seen his dad, to have returned home again.
When I was a child, I walked home from school, and instead of ringing the doorbell, I went around to the lounge window to see if mum was in there and to wave. The lounge was a long room with French windows at the far end. I could see mum kneeling on the floor looking at something like a photo album in front of the French windows. She didn't notice me waving. So I went round the back of the house, intending to knock on that window instead. By the time I'd got there, she'd left and was in the kitchen, so I knocked on that window instead and she let me in. She asked if I'd rung the doorbell and she hadn't heard (was very hard to hear the doorbell in the kitchen at the back of the house) , and I replied that no, I'd just seen her looking at something on the floor of the lounge a few seconds ago. She told me she hadn't been in there at all, and had been in the kitchen the whole time.
I never believed I'd seen a ghost. What I believe (if not coincidence) was that I knew telepathically that she was at the back of the house and wouldn't hear the bell, and that's the way my brain relayed the information to me. Same with my grandpa's story perhaps.