The ground braid is a safety ground, and usually not a part of a circuit ground. A pinball machine can operate fine without the ground braid, it's purpose is to make sure that you don't become a part of the ground path for high voltage in the event of a short circuit.
A ground loop is when there are multiple paths to ground and they are not all at the same potential. Ground loops can be anywhere in the system, not just the same connector Problems such as noise in audio systems are caused by ground loops, as noise immunity in amplifier inputs is compromised. Star grounding is one method to combat ground loops. Star grounding is connecting all circuit grounds to the same point, so that they are all at the same electrical potential.