Hi - i am not an expert, i want to say first off -- I am a student. But, I do feel there is a simple explanation for you:
The pirouette from 5th (example, Right foot in front, in 5th position), turn to the right lifting Right leg to Left knee, landing the turn with the Right foot ending in back... This is a pirouette en de hors. This is, following your line of thought, the first pirouette from 5th i would say, that is taught. An important point, too is that the preparation for this pirouette en de hors from 5th is Right arm in first, curved in front of you (I apologize if my terminology is not 100% correct, but am fairly confident that this information though, is correct), and the Left arm is out to the Left side in 2nd.
Starting in that same position of the feet - Right foot in front, 5th position, Left foot in back, turning to the Left, lifting the Left leg to the Right knee is an en de dans turn. The preparation for this en de dans pirouette from 5th is Left arm in front of you in 1st; Right arm out to the Right side in 2nd.
In other words both ways are "proper" as your note above asks. They are simply different turns - one en de hors, the other en de dans, which determines the arm positions, direction you turn in, which leg is the working leg, which is the supporting, and which foot lands in front or back.
Please don't feel "crazy" - the en de hors vs. en de dans turn concept just takes time and considerable experience to become very familiar with but I see you are on the right track, since you realize that indeed you have seen, and there appear to be, two categories of pirouettes from 5th - both correct -- just different!!
I hope others will comment and perhaps clarify better, especially the many experts here.
If I as a student am not allowed to make these comments here, please feel free to delete or disregard! (Needless to say if I am incorrect I'd be very interested in knowing where I've gone wrong...) --