Pretty much everyone knows that the strap between the bridle and the rider's hands is called the "rein" or "reins." (It's plural even though some English reins are a single strap).
So, what are the lines?
When driving, you don't call the strap between your hands and the bit the "reins." You call it the lines - perhaps due to its much greater length. (This is a common mistake made by writers).