@TreadSoftlyOnMyDreams Your well-meaning advice needs a bit of qualification and is probably not going to be relevent for the OP on her first trip.
Putting the rescue number on speed dial is good advice, but just as important is knowing what to do in the event of an accident, and knowing where you are on the mountain. It also assumes that you and/or your children can speak the language of the person on the other end of the emergency number.
The best action to take is to stop the first adults who come by (many will stop anyway) or even better, stop any ski instructor or piste patrol worker. Instructors all carry basic first aid kits and are trained in first aid, have direct dial numbers to the rescue team, and often also have radios and can contact colleagues nearby. Someone (i.e. an adult) should also be told to ski to the nearest lift to report the accident - you might not be able to get a mobile phone signal. That person should also then take the lift back up and ski down to you so that you know a) that the accident has been reported and b) what is happening next and when.
Always carry a piste map so you know where you are - pistes are also marked by numbered "lollipop" markers and in many resorts the emergency number is on the reverse side of these markers.
Nobody will be left on the mountain just because they don't have proof of insurance (what would happen if the skier is unconscious?) but having the details and contact card etc. on you is excellent advice and will speed things up throughout the process (rescue/transport/clinic/hospital) as well as making it clear who is going to pay.
Modern chair lifts do not usually transport skiers down the mountain, and should not be relied upon as a method of getting down should you have "misjudged a run." You might be able to get a ride down on a rescue snowmobile if you are very lucky and are willing to pay a rescue fee (min. €150 where I ski) but really don't go anywhere that you're not sure of being able to get down. The rescue service is for evacuating injured skiers and emergency cases - it is not a rescue service for people who over-estimate their abilities.
The final advice is perfect - do not ski if you're feeling tired and err on the side of caution if the run back to the village is crowded and/or icy at the end of the day. If there is a cable car or gondola available then take it and save your knees (and confidence) for the next day. Currently conditions in the Alps are very warm and the pistes are slushy from midday onwards - after lunch sit on a terrace in the sun, have a hot chocolate and ride down if at all possible.