I believe that "remains part of the bat" refers to devices used during warmup, and certainly must come off before going to the plate with bat. Normally, batters warmup by holding the handle end
so as long as the device does not come off while swinging warmups it is legal. The "warmup bat" shown in original post is a weighted bat made for the purpose of warming up. It has a handle with a knob on it, like a regular bat, and a weight at the other end. It does not have a barrel because there is no need for one.
A doughnut that flies off the end of the bat during normal use is illegal.