A lot of people discuss having more weight in the outside rein. However, my coach wants equal (light)weight in both reins with the horse balanced and carrying himself in the middle. The other day as I was cantering I miraculously got a collected canter with the perfect light balance between both reins and my horse did what I will call a perfect line of tempi changes - straight, expressive, calm and balanced. I hope I will be able to repeat that! Now I understand how much better it is than having more weight in the outside. YET so much time is spent telling riders "inside leg to outside rein". I think that is misleading and a misconception on the part of the coach.
I was always taught that the outside rein is for the horse and the inside rein is for the rider which translates as the default is the outside rein unless you are doing something such as a half halt when a vibrating inside rein may be required to get attention from the horse. For me there is always more weight on the outside rein. My horse has been taught this from day one because I broke him and that is all he knows. With a green horse both reins will have equal weight but then I switch to more outside rein over a period of time. This method prevents the horse from leaning which develops into pulling if unchecked. When I feel the lean then I add more inside leg to push him to the outside rein. My eventuall goal is to have the horse move off the pressure of the outside rein (better known as neck reining). It is currently unfashionable to call it this! But in the past dressage/war horses learned this. Now that my horse is learning collection I ask for the collection with the outside rein(by making it shorter) and my seat.
I just noticed a video of a freestyle on the main page done with no bridle just a rope around the neck. Yes I use this as well along with my regular bridle. This is a part of TTEAM and is called a balance rein. Riding with this causes you to rethink the need for an inside rein. It acts like a big outside rein which is what you need the most.
I think I know what you mean by saying that you have equal weight in both reins. I think that's the ideal and the correct way to ride. I think the idea of pushing the horse into your outside rein from your inside leg doesn't mean that you have more weight in the outside rein. Your reins have different functions. Your inside is used for flexion and the outside is used to prevent too much bend and to turn the shoulder. When you're doing things like tempi changes the difference in weight in your hands would be much more subtle and the softening you would perform when your horse flexes in the jaw and poll as you're changing leads would be very slight but I'm guessing it would still be there. There are some really good videos of Hubertus Schmitt explaining how to ride your horse from the inside leg to outside rein and he makes it sound and look really simple. With a green horse he basically bends the horse's neck to the inside with his inside rein until he feels it take a hold of the outside rein while energizing the horse with his inside leg and as soon as the horse softens, he softens his inside hand very slightly while still maintaining the contact. As you move up the levels, these functions become more refined as to be virtually invisible and very subtle.
Initially, the horse should be evenly into BOTH REINS (even though the horse is 'positioned inside'), only then will the horse take a correct hh/fold the hind legs. As the horse develops more self carriage, becomes more laterally flexible, reacts with its inside hindleg (from the riders inside leg), the horse will then 'stand on the outside rein' (between the inside leg and the outside rein). In shoulder fore the horse is supported by both reins lightly, in shoulder in the rider should be able to give the inside rein and the horse will stay 'in front of the leg' and into the outside rein. Merely lifting the inside rein slightly should indicate a change of flexion from the action on the corners of the lips (not more weight per se).
The horse FLEXES laterally at the atlas axis because of the inside rein, it BENDS evenly through the body because of the (placement of/reaction to) the legs. With tempi changes is a very small change of flexion (mm), not change of weight in the hand.
The riding of a horse in a neckstrap is merely something that can be played with, it is not the basis of training. That said the flexion on a circle/in lateral work IS necessary if we are to get to straightness/control of the outside shoulder (since the horse is trapezoidal in shape).
If you use the balance rein as the basis of training your training job becomes much easier. Your horse is balanced and later on the balance rein is used to teach collection so that the horse will lift its back and carry. It is also very useful for improving transitions, at any level. It is also very helpful with horses that pull.
For a beginning rider, or a horse whose mouth has been abused, a stirrup leather can be used as a neck rope. (It is something that many young kids do, and is impressive to viewers who think riding w/o a bridle is difficult (it actually gives a lot of control). It however would have nothing to do with collection (amplitude of stride into shorter/higher steps). People pull, horses only react to what riders cause.