Hell may have frozen over because I agree with Dot! And I acknowledge she has way more expertise in this area than I do.
The walk is both videos of the first horse is not lateral. It has a clear 4 beats and shows a clear V. While the rhythm may not be perfectly regular, we don’t see anything resembling the two right or the two left legs moving together, or even close together, which would be a lateral tendency.
The TWH is an excellent example of a lateral walk