post a photo looking into this overflowing tank please. other pics showing all the pipes connected to it, and the one hanging over the tank.
use a piece of string to tie up the ball float, which should prevent water entering through the valve. Run the tap that you said prevented the overflow, for a couple of minutes.
go and take another pic of the inside of the tank, showing the water level. Is there any sign of water entering it?
Wait ten minutes. Photograph the inside again. Has the water level changed?
Take another showing both the tanks, and indicating which is higher.
What colour is your hot-water cylinder?