The function of the laps is diferent looking both vídeos.
The first lap is for external diameter.
The external piece (part with holes) distribute evenly the stress like you said. And the internal piece (tubing with a cut) redistributes the stress more evenly (in the vídeo he said the helical cut is better than a straight cut because the straight can get oval).
The way the geometry works is similar to bicicle wheels. The hub and rim are conected by the spokes and remain centered.
This lap works exactly like this. Using the screw in the external part will change the diameter of the rim, the spokes remain with the same distance, so the iner part (hub) need to acomodate the new geometry. The spokes apply exactly the same force, so the inner part adjusts equaly.
The second lap is for internal diameter (this one I think the geometry is not perfect).
When you apply stress evenly the outer ring will open (looks like not a perfect round part, but definetly closer).
In 15:40 of the vídeo he says that lapping like he's doing the position of the piece need to change time from time to compensate if geometry is not perfect.
You need to remember that laping is used to very precise and minimal changes in diameter, so the diference in tool geometry will be closely to zero.