The way the heddle works, like any type of tabby weave, is every other warp thread goes through the hole in the heddle, the wood device, then the other warp thread goes through the slot next to the slat with the hole, when you lift the heddle the warp going through the holes move above the threads in the slot, when you push the heddle down the threads through the hole go below the threads in the slot. This is called a simple tabby weave.
So when beading you place the heddle threads above the stationary warp threads, lay your strung beads down and spaced each bead between two of the stationary warp threads, move the heddle down, reversing the shed, this is what the opening of threads created by the heddle moving up and down and called. With the heddle now down, string your nex course of beads as above then lift the heddle up. This is the purpose of having two warp threads between each bead, as shown in the diagram at that web site. With this type of beading you would want to weave a few threads with out strung beads, to start the weave, before beading.
By stringing the beads, laying them, reversing the heddle, again laying the beads, with each reverse of the heddle and course of beads you are locking each course of beads. It seems easier to me than the way we are doing it now, having to actually string the course of beads twice.
I hope what I wrote makes sense.