Quess I miss judged your guestion. Thought you had a hidden statement in there! Your right, that catching an edge excuse won't fly with your instructor, if he possess a keen eye he will know the exact technical reason for your fall and will only ask you to see if you do also.
Over the years I have heard "catching an edge" used as an excuse for a multitude of technical sins. The majority of people use the term as a blanket excuse for falls of which they don't understand the cause. Usually there is a tehnical reason for the fall that just saying "I caught an edge" does not address at all.
The only instance which I feel the term accurately desccribes the cause of a fall is when the skier is purposely trying to ride a flat ski (not turning). When that is done the skis have a tendancy to swim back and forth. In doing so it is possible that one of the edges on the tips (especially if not properly detuned) could catch an unevenness on the slope and engage. Falls resulting from this can be rather ugly because speeds are ussally high (not turning) and it results in either skis crossing or skis going spread eagle. Hurts just to think about, don't it!
Most people, at most times, are not riding flat skis however, they are linking turns, so "catching an edge is not an accurate explanation for a fall. If you are using your edge to turn you can't catch it because your suppose to be on it. It's not like it has hooks on it and its going to get caught on something and get ripped out from underneath your body. If it shoots off in an unintended direction while you are executing a turn it's because of inappropriate edge ussage or poor balance and pressure control, not because a little snow snake grabbed it.
Hope that better address your question.
As for inside/outside ski/edge, thats pretty easy to explain. From the start of a turn to the finish of that turn the outside ski is the ski on the exterior side of the arc. In a left turn your right ski is the outside ski and your left ski is your inside ski, all the way through the turn. The inside edge of either ski is the edge under the big toe side of your foot. The outside edge is under the little toe side of your foot. In normal carved turns the majority of pressure will be on the inside edge of the outside ski. When the inside ski is pressured in a turn that pressure is directed to the outside edge of that ski. When we lift the outside ski and execute a turn on the inside ski all pressure is directed to the outside edge of that ski.
I think we should just throw the uphill and downhill ski terms right out the window because they just cause confusion. They're only usefull in differentiating skis while in a traverse.
Hope that helps.