Stem christie is a step or slide of the outside ski. Eg. to go right you are stepping the left ski out and at an angle with tip towards the right. You can also slide it on the snow.
What you're doing is instantly establishing a steering angle. Your outside ski is now at an angle to your travel direction and will start changing your path when there's some weight on it. You then match the inside ski to it. If the stem is small you can steer the inside ski parallel to the outside.
Basically it cuts the top of the turn off. It's actually quite useful in trees. I actually teach it for that. ( this is trees with packed snow. Stemming in powder could be problematic without a monster Fatypus ski)
You're close to the tree going across the hill. Step out your left ski into a turn and match the inside. You're now turning around the tree.
So "Christie" comes from Christiana. A "town" or region in Norway that later became Oslo.
Like Telemark, of telemark turn, is a region in Norway. Both types were developed by Sondre Norheim I think.
Christiana turn would have been the early parallel (ish) turn as opposed to the telemark style turn. Don't forget there were no bindings, you used leather thongs to strap the ski on, and boots were short leather.
Stem christie used to be part of a learning progression to parallel. It has not been so for at least two decades and probably 3 but
@Bob Barnes could tell you.
A wedge christie is steering the inside ski to match the outside ski. To do it " correctly" the tips of skis open - you steer the inside in direction of turn, instead of closing the tails to match the skis. It demonstrates independent leg steering.
By far the biggest problem with demoing wedges and wedge christies is excessive speed. Most go way too fast. You can actually demo a wedgie christie by trying to do a slow parallel turn. Bring the turn uphill a little slmost to a stall. ( stop) When you start the new turn you will naturally make a wedge. Again, these are slow. I'd estimate that 7/10 instructors not recently in an exam training or demoing environment, will go too fast.
( I've seen it in an exam- even by an examiner too when not testing for the maneuver specifically)
So slow way down! Use line to control speed. Just doing that will set you up for good turns. Excessive speed will cause many, many problems from backseat to closing to match to just skiing parallel. Also too steep terrain will cause weird things.
Here's
@Bob Barnes illustration of matching versus closing. You want matching for wedge christie. Guide the inside tip around the turn. ( guide both actually)