[GroundUrMast]

The link you cite offers an excellent example drawing in the section titled "The NEC requirement". There are two ground rods shown. The ground rods are "bonded" together with #6 AWG CU. If that bond were omitted, the ground rod under the antenna would have a connection to the electrical service ground only through earth (or possibly the tuner in your TV). The real world impedance/resistance a lone ground rod can be anywhere from a few ohms to hundreds of ohms.

Voltage = Current X Resistance. If a fault current flows in an isolated ground rod, voltage will develop. The #6 CU wire can be depended upon to provide a predictable low impedance connection, whereas earth can not. By bonding all parts of the grounding system together with low impedance conductors, the voltage difference between grounded objects (such as the AC neutral conductor powering your television and the antenna coax) will be held to a minimum if fault current flows somewhere in the system.

A difference in voltage will not develop on it's own. An outside energy source is required. That can be from atmospheric static, or from insulation failing - allowing an AC conductor to contact something that should be at ground potential.

The longer a conductor is, the more resistance it will have. #6 CU wire has about 0.4 Ohms per 1000 feet. Going around bends will add impedance at frequencies above DC, but at DC and power line frequencies the increase is negligible.

I very much agree that the grounding wire from the mast, to the the connection to the ground rod, should be as short and straight as possible. If you take a direct hit by lighting, you want to make the path to earth as easy and inviting as possible. Once lighting has found it's way to the ground, stand back and let it go where it wants to.

As mtownsend observed earlier, if lighting is a significant risk in your area, a dedicated lighting rod is something to consider.