Bottle dungeons, also known as Oubliettes (from the French "oublier" meaning "to forget") were commonplace in medieval castles. Prisoners were thrown into the inverted bottle-shaped cells to be forgotten, hence the names. People could easily be thrown in, but were difficult to get out. Mostly prisoners in oubliettes would die of thirst or starvation.
The bottle dungeon shown in the first episode IS the one in the Castle grounds. They clearly cross the wooden bridge over the moat to enter. It's barred, for sure, but that's the Castle.
The large open parallel shaft one of the students is dropped into is not a Bottle Dungeon in the Cathedral grounds. The shaft shown is huge and no such large open shaft exists in the Cathedral grounds. There are one, maybe two, small wells coved in wrought iron protective lattice covers. The Bottle Dungeon referred to is in St. Andrews Castle, is much smaller than shown, and is unsurprisingly, bottle shaped.