Well...
A potentiometer is equivalent to two variable resistors in series, with a constant total resistance and a variable proportion. The connection point (the middle pin) has a voltage between the voltage levels applied to the end of the POT (commonly a GND and a Vcc, but not always).
Since the total resistance of the POT is fixed the current is dependent on the total voltage applied to the POT, in cases like the demo the voltage is constant too (by design), so the current is constant, unless there is comparable drain on the 'middle' pin.
POT is normally used to provide a reference voltage, which is MEASURED between the middle pin of the POT and either end pin (commonly the one connected to GND).
Another method is to use the POT in series with a fixed resistance to provide CURRENT control.