Is the diameter of drain snake cable the bigger the better?
Bigger is better to a point. The problem comes when the size is two large to make a 90 or 2. If the cable is two large in lets say a 3" pipe the cable can chew through the side wall (I did this with a power snake) I find the smaller cable with a center core tends to hold up better and make it through more bends. You did mention "machine" On smaller hand snakes I have never found one to be two large If this is the type you are talking about. I use smaller hand style like a drill with snake drum for laterals and my big snake for main line work where I enter through a clean out or vent. A small snake can get twisted up in a large pipe when the clog is encountered.