Actually, those are referred to as "virtual terminals". "Terminals" without qualification refers to the old-time time-sharing setup where you actually connected multiple physical teletype (or glass TTY) devices to the system. It also refers to generic terminal sessions including SSH sessions.
In Red Hat, you can get to the first few virtual terminals even while running X, but you have to use CTL+ALT+Fn to get out of X and into the virtual terminal space. X continues to run, however. I think it's bound to ALT+F7 (I can never remember, so I just push buttons around that part of the keyboard ).
"privilege" comes from the Latin words for "private" and "law" (legal) and dates to feudal times. To "claim privilege" meant that you were above the laws that applied to the common people.