Good news about the neighbour having spotted Shai. . I apologize if I seemed accusatory when I posted about the risks for an outdoor cat, I was only typing thoughts that crossed my mind. Sometimes words only don't convey true feelings.
From your description, something may have spooked your cat and caused him to act out of character. A mouse, a squirrel, even another cat on HIS property may have caused him to bolt outside in spite of the rain.
About the food, if you put any out for him, don't leave it sitting there, only put it when you're around. It may attract other animals even cats, and there could be territorial issues that would make it harder for Shai.
I hope you get your kitty back soon. Those days and nights of waiting, hoping and looking are hell, I know, but don't give up hope. I once had a cat, a semi-feral rescued cat, who bolted out and was gone for 3 weeks. She came home though, and we had almost stopped hoping. We figured that she must have remembered home was more comfortable and the food was more regular. Sending you and Shai lots of mojo.