I hate this expression, to me it just conjures up images of horses where attention is fixed on the unnatural head carriage, by pulling to a perceived position of what conformation should be. When we should be looking for overall balance and harmony of the rider with the horse, not just the results of someone hanging off the reins. Pulling a vertical head position has nothing to do with collection. It has the opposite effect preventing engagement and creating an insensitive, unresponsive horse.
When training a young horse we need to maintain a sensitive and soft mouth. Harshness either inflicted by the trainer, or in self- defense from the horse, will cause harm, a piece of metal in the mouth which is pulled and tugged on in all directions is not a treat for anyone, let alone a horse.
A sensitive mouth, for obvious reasons has to be a primary objective.
We really need to be using words like connection, softness, balance, engagement, lightness, softness to name but a few, not just taking this expression as part of the horsey vernacular …. Use it lightly .. Lol