It really depends on what kind of reins you have. Either way, riding western on a neck reined horse, they should be held in just one hand. If you have single piece or sport reins, then just wrap your hand around them and hold them.
If you have split reins, then you can either criss-cross them over the horse's neck and hold them the same way you would a sport rein or keep them both to one side.
If you bridge them over the horse's neck, then they should be held like this. This is also the same way you would want to hold sport reins.
If you opt for holding them off to one side, the proper way to hold them is like this in your left hand with the slack hanging down the left side of your horse.
However, if the horse is not trained to neck rein, then you would want to hold the reins similar to how the English riders do, only with more slack.
If you took that photo of smrobs on her horse and photoshopped her onto a horse with an English saddle, she would be sitting nearly identical. She might have her legs a tiny bit shorter, and she would lower her hands and have one on each rein, but other thatn that, the correct position for sitting ahorse is universal.