I just wish to add:
I am aware and have seen first hand the damage rope halters can do simply by using them for training or leading purposes. Tana, an arab broodmare got skinned on her nose from a lunging session because I kept having to bump her nose in because she was leaning on the line and not paying attention.
I have also ridden in a halter, and a rope halter. Neither offer me the same "messages" as my aeron mack riding halter. I feel my messges are disgruntled by the fiador knot of a rope halter, and on nylon, I simply do not have the fixed messages that the knots a rope halter have. On both, the pressure was applied broadly on the nose, I felt, and I have had no crisp signal.
Again, I prefer riding in my riding halter, but doing so is rare as 90% of the time, the horses I ride are bitted for maximum signal sending and safety for myself. As much as I deteste riding bareback in a bit (don't ask me why, as there really is no real reason behind it) I will do so if safety demands of it.