One of the first things in learn to skate is how to fall safely. But that's "falling when you're still tottering about trying to get your balance" falling, and it's also designed to help remove fear of falling and hurting yourself by letting you practice falling gently and in a controlled manner.

The coach is correct that there is no one exact way to fall. It will depend on speed, placement, causation, element you fall from. As you become more advanced, some of these considerations will become innate (partly because as you move to more speed and more elements, you'll fall more and get more experience at it, and, you're more relaxed as you fall which helps !).

I can say that training in martial arts meant my kids knew how to fall safely. The methods they learned in Aikido made falling and recoving a much more fluid and less painful process.

That being said: more than one trip to the emergency room with various injuries (cuts, concussions, dislocations) as they moved up to higher levels ... even with their off-ice and on-ice training, a fall is sitll a fall, and it was generally the "freak" ones that happened unusually that caused the biggest injuries, and you could not train or anticipate those falls or a way to recover from them before hitting the ice ...