The pivots in the calculated row echelon form are after the 0s in the diagonal, which is contradictory to the definition above. I am a bit confused about which one is correct.

I agree that those two contradict each other. He must have misspoke in the first video you linked. The leading 1s in each row of a REF matrix need not be on the main diagonal. As a matter of fact, the only times all the leading 1s are on the main diagonal is when the original matrix is full rank (non-singular).

Iām really not sure what he was trying to get at when saying that the numbers after the 0 in the main diagonal need to all be 0s. The numbers BEFORE the pivots (leading 1s) in each row, however, DO need to all be 0s. Also, there can be at most one pivot per column in a REF matrix.