When I was a teenager ca 1950 were indeed told to stand in our stirrups to canter. I had the occasional ride, a lead rein hack, as a treat. My horse was led by the (mounted) teacher holding a lead rein and yes we cantered.
When I learned to ride 50 years later with proper lessons instead of canter being so easy, it was regarded by everyone as an enormous mile stone and we had to sit to canter before were were allowed to hack. Learning how to ride forward seat was now a considerable problem, involving bending of one's body and legs at the correct angle and balancing one's head going forward and one's behind going back.
It wasnt till I heard Tik Maynard explain jumping position that It was simplified into a discussion of how one's legs should drape the side of the horse and in effect that meant standing in one's stirrups with bent knees. Which I can still do to perfection.
The aim of this OP is to preserve your back and also the back of the horse. By the time you get to old age, sitting for long fast canters is likely to wreck your back. But in UK it was always a RS exercise to ride any gait standing in one's stirrups. Even in walk it is not that easy even for experienced riders who have not tried to do it for years.
But your weight should not be going forward. The stirrups are still on the saddle, near the girth and your weigh should be central (front to back) . If you put your weight forward and the horse stops suddenly you will fall off forwards and that seriously can be lethal. Even in walk.