Why would he call dead ball, it's allowed according to what you've been quoting, so should be given out.
"If the non-striker is out of his/her ground from the moment the ball comes into play
to the instant when the bowler would normally have been expected to release the ball, the bowler is permitted to attempt to run him/her out"
They have to have the bails off BEFORE the moment they would normally have been expected to release the ball. Obviously if they go through their bowling action, that point has passed. In fact, given that it takes about half a second to spin round and get the ball down to the stumps, realistically the bowler has missed his chance by the time his arm comes above his shoulder.
and yes, this does mean that by the letter of the law some of those umpiring decisions in the clips at the start of the thread are just plain wrong.