The riders on the lipizzaners aren't keeping a tight rein on their horses. If you look closely in the photos, the curb rein doesn't even seem to be engaged. The horse is ridden on a direct contact with the poll more elevated than a western horse. That's just a difference in discipline.
The ideal dressage and the ideal reining horse have similar goals. A lightness of aids with the horse using their body correctly.
Although the dressage horse is ridden with direct contact with the bit, the communication comes through their other aids like their weight, seat, and legs.