Lets start on what is verticle. If I have a peice of wood thick enought to stand on its own.
Scenario.1. To move. The whole lot is moved as a unit.It can lean to its right and return to its original position it can Sway to the left and return . It can do this whilst moving.
Scenario 2. The top of the peice of wood is tilted forward onto its edge .which is edge one. Now the the base which is moving must catch up.But the top has moved again. Now the base must try to catch up again. It becomes a bit like a dog chasing its tail. Except the peice of wood never stops falling
Scenario 3. The peice of wood is bent over at the top. It will never stand on its own. This is the type of dancer who has never stood up straight since they were a child. So they adjust their steps to suit their posture. This is a typical older generation Social Dancer.It will never fit into the competition scene. Unless the whole field have the same problem.
Lastly. I saw this couple dancing socialy. He had been a competitor many years before and has been teaching ever since. They were doing the Rope Spin in the Rumba ( for those who don't know the lady circles the man). This guy leaned backwards much further than most would be able as the lady went around This is an example of what can happen creep in if allowed. It didn't look good.

Quite obviously, NONE of your scenarious describe what a skilled dancer actually does.

What you've left out (except in the broken case 3) is that dancers do not move as rigid bodies.

Given how much you like to remind people to use their knees, it's really silly for you to try to explain movement or verticality while leaving them (and the hip joints) out!

Put knees on the peice of wood and what do we get. Is it still catch up time or not catch up time. Are my feet trying to catch up to the rest of the body. Either they are are they are not. There is no inbetween.

"Put knees on the peice of wood and what do we get. Is it still catch up time or not catch up time. Are my feet trying to catch up to the rest of the body. Either they are are they are not. There is no inbetween."

At times they are. At times they aren't.

What you miss by leaving out the knees and hips is that they enable the body to be in front of the standing leg, or behind it, while remaining vertical.

Your board can't do that. It doesn't have joints, so the whole thing has to lean. Thus it cannot execute anything comparable to the action of a dancer during step 1, when the use of these joints lets the dancer keep her body vertical, even while the location of the center relative to the feet is changing.

So sometimes I am falling and some times I am not falling. Does my partner know this. Does she know when you are going to propel yourself at her. Or when you are not. Tell me about the ladies half of this equation. Where exactly is her body in relation to her feet. Just on one step will do. Is it a mirror image of your man. If it is she will be getting close to falling away, don't you think.