The turn coordinator, but also its predecessor (however you want to call it) use the principle of precession to indicate the direction of the turn. To be able to use the principle of rigidity, the gyro will need to be able to move within its mounting, around all three axis. The gyro in a turn coordinator or indicator can only move around one axis, thereby creating a sensor that picks up movement around the vertical axis of the aircraft. It does this by exploiting precession, as a turning motion around the vertical axis will precess and cause the gyro to cant left or right around the gimbal axis.
What this question is trying to do is make you think about the principles involved, not just whether there is a gyro within the instrument.
As for the terminology, you can have either a turn coordinator or a turn & slip indicator in your aircraft. These names refer to the entire instrument, which incorporates two parts:
1. A slip indicator, which is the curved tube you mentioned.
2. A turn indicator, which uses the gyro as I explained.
In a turn & slip indicator, the gimbal axis is mounted horizontally, while in a turn coordinator it is tilted rearward by 30 degrees. Because of this the turn coordinator will start to indicate a turn once the aircraft starts to roll (the tilted axis causes it to sense direction around the roll axis as well, although the movement around the yaw axis will overrule this once it gets going), whereas the turn & slip indicator will not start to indicate the turn until the aircraft is banked and starting to change direction. Because of this 'delay', turn & slip indicators can cause the unwary instrument pilot to overbank, as he/she will not see the instrument react to the initial movement. Use of an artificial horizon together with the turn & slip indicator can alleviate this problem, but I'm probably digressing....
Edit: the Wikipedia page about these instruments is actually pretty good, see here:
https://en.wikipedia.org/wiki/Turn_and_slip_indicator