The pull-up training pants are the step to underwear, and are underwear in the child's eyes, not nappies. The child learns to go to the potty and pull down the pants before sitting down.
In relation to the hopping around etc, it is the child learning to react to the stretch sensor in her bladder. When the bladder fills, it starts to stretch. This triggers stretch sensors, closes external sphincter and opens internal sphincter. The closure of the external sphincter needs focus, and allows the bladder to stretch. When the bladder stretches a little, the internal pressure sensors tells the child she does not need to void, so the internal sphincter closes and the external sphincter opens. As a result, the need to void is gone. Urine is still entering the bladder, and the stretch sensors retrigger, the internal sphincter opens and the external sphincter closes.
This loop continues until the child makes the decision to void OR looses the battle in holding her external sphincter closed.
This is required in order for the bladder to stretch and give the child night-time reserve - night time control is based on the quantity of a hormone that reabsorbs water in the kidneys, and the capacity of the bladder.
The hopping around is part of the steps the child needs to do in order to get bladder control. There is a key moment that the bladder can't expand any further and the external sphincter can't hold back - and that is the level the child needs to know about which needs focus.