It's a combination of things.
American style uses a slightly different technique than Int'l, although there are still several schools of thought in terms of what constitutes "correct" American style technique. But most people agree that there is at least some difference, so it doesn't really make sense to to quote Laird when discussing American style.
On the other hand...
If Laird really wants you to have only one hip change over three steps, and doesn't think the feet should fully close on the second "cha", then this technique is highly outdated, even for Int'l style. That would be considered extremely sloppy by today's standards. If you truly leave your hips in the same position for all 3 steps, it will probably look like bad EC Swing.
As for music, yes, that will always have some impact on the quality of movement, partly because of the tempo, and partly because dancers react differently when they hear music. But none of these factors should prevent foot closure or 3 hip actions over the three steps.
Latin hip action, by the way, generally isn't lateral. Hips rotate around the spine. It may appear to be lateral when the action is smaller, much in the same way a small section of a large circle may appear to be straight. So what you're seeing is a lesser action on counts "4&", which I would consider correct. Since counts "4&" last only a half-beat each, there is less time to complete the action.
Regards,
Jonathan