Storage on site vs in warehouse

Hello all,

I'm new to the game and am trying to figure out storage. I have my woodcutter up and running and fully staffed with 5 workers. I also have a lot of on-site storage in my woodcutter as well as two small warehouses with wood storage.

I wanted to see if I could only have wood stored at my woodcutter so I removed all wood storage from my warehouses. My guys moved wood over to the woodcutter, but the amount of wood in my storage menu on the right-side of my screen did not go up with the wood taken from the warehouse to the woodcutter storage.

How does storage for the woodcutter (and other workshops) work? Do I want products stored there? Does it count towards the storage menu?

Thanks!