To clarify, with shielded power cables there are up to 3 ground conductors. A foil, a small drain wire, and a equipment grounding conductor which is the same gauge as the hot or neutral.
If the shield is braided there is no separate drain wire.
In case of a foil shield the foil is nestled against the drain wire. The drain wire is there specifically to make contact with the ground pin on the plug. You don’t actually connect a foil, when present, directly to that pin. It is this drain wire which you only connect at the wall side, and leave the equipment side unconnected.
The EGC, which is insulated and green gets connected at BOTH ENDS.
In the event of an equipment short from hot to chasis, the EGC alone carries that current.