After being around grounded workstations for decades now, I'd say you've got it down correctly. The black wire connects the mat to earth ground and the mat is conductive to dissipate any static charge. Plugging in the wrist strap grounds you so nothing you touch will get shocked, rendering you harmless with regard to static.
And nope, doesn't matter which plug you use; there are dual receptacles to allow two people to work at the same time.