The wedge/key fits through the barrel tenon, and stock escutcheon, for the sole purpose of securely holding the barrel in the stock. If it is not secure/tight, then you must peen the tenon slightly to provide a tight fit. Now, as to direction of fitment, it is personal preference. I push mine in from the left, because the smooth head or the wedge does not gouge my palm.
Bottom line...
Do it as you like, since it makes no difference.
Bottom line...
Do it as you like, since it makes no difference.