I have taken lessons recently with two different trainers, With one trainer, when I am trying to turn right she has me put my right hand forward, and to the side. The other trainer has me pull back with my right hand.
When you move your rein out to the side that is called a leading rein and when you move your rein back towards you (while giving with the outside rein) that is called a direct rein. Generally, a direct rein will make the horse bend his body into the turn which will give you a more precise turn than a leading rein.
Both are NOT correct, both are rein effects however. A direct rein of opposition Pulling one rein is a backward action, it will cause the horse to throw the quarters out and (over)bend the neck, and if combined with letting the other loose only pulls the bit through the mouth (unless there is a D/fulmer/fullcheek). IF the student learns to pull back with the reins it is very difficult to overcome/replace. The horse must come to the contact, not the opposite.
When using an opening or leading rein, the rein comes lighty away from the neck (NOT forward), or in a more subtle way by merely turning the thumb over to the outside (ie nails showing/little finger coming closer to the neck). The purpose of such a rein is to direct the horse into the outside connection which then allows for straightness.
That said, the outside rein MUST be kept lightly connected, it does not go forward/release, it functions as the 'receiver' and forms a kind of wall that the flexion inside can move into. Both reins must have the same weight (lightly/ounces).
Sorry, the person above is correct in the sense that the leading rein does not go forward. My comment that "both are correct" was in relation to a leading and direct rein being acceptable methods to turn.
Here is a screenshot from one of my books showing the difference:
Both ARE rein effects (for teaching cavalry officiers initially), but pulling one and letting go the other is a method which destabilize the bit's center. (The good part of the Chamberlin drawings is that the horse up/open, if it were a closed posture the direct rein of opposition would be onto the bars.