“How long will you be staying in Japan?”
“Please stay here and wait. I’ll be back in five minutes.”
“Use an umbrella to stay dry.”

“Stay” means “remain the same”. While you stay in Japan, or stay inside the house, your location doesn’t change. Then when you leave Japan or go outside the house, your location changes. If you are not wet, and you use an umbrella so that you don’t get wet, you remain dry, and your state doesn’t change.