If White captures a pawn he can still capture the other pawn if he is able to place his King inside the square as you have seen in the lesson about the square rule.

During a normal (non internet) lesson it is rather easy to indicate the squares, but I think that you are right that it is not so easy to see them in the lesson above.

It should probably be stated that the white king in this example must be inside the overlapping promotion squares of the two pawns.

There are other hidden assumptions too. The pawns must be level. The king is on the 8th rank in all examples. Does it need to be for these examples to work?

Another hidden assumption is that black wins only if it is white’s move, or that white is forced to move the king but black is not forced to move its pawns. The hardest part for me is imagining how the white king can prevent any separated pawns from promoting regardless of what rank they are on. It seems if white captures on pawn, the other has already escaped towards promotion.