Electrical – the difference between a potential divider and a potentiometer

The correct term for the common terminal of a potentiometer is the slider.

A rheostat is simply a variable resistance used to control power to a load and you are correct about the wiring. Only the slider and one other terminal are used.

A potentiometer uses all three terminals, enabling a variable voltage or signal to be tapped off from the slider.

Potentiometers and rheostats are made the same way, but rheostats are usually much "beefier", as they are generally used in high-power situations.

The slider is often connected to one or other terminal for safety reasons, in case it loses contact with the track.

In general, voltage division applies to the voltage across a resistor in series with one or more other resistors.

For example, assume that we have \$N\$ series connected resistors and, further, that the voltage across the series combination is \$V_S\$.

Then, the voltage across the \$n^{th}\$ resistor is given by the general voltage division formula:

$$V_{R_n} = V_S \frac{R_n}{R_1 + R_2 +\; ... \; + R_{N-1} + R_N} = $$

In the diagram you provide, it happens to be the case that

$$V_{out} = V_{R_2}$$

And, by the general voltage division formula applied to two series connected resistors

$$V_{R_2} = V_{in} \frac{R_2}{R_1 + R_2}$$

Thus,

$$V_{out} = V_{in} \frac{R_2}{R_1 + R_2}$$

Note that we could have chosen \$V_{out}\$ to be the voltage across \$R_1\$ instead though this isn't often the case.

In most cases, the output voltage is referenced to ground and, so, the output voltage is taken across the resistor with one terminal grounded.

In summary, voltage division is more general than the two resistor voltage divider circuit you've provided. The principle of voltage division and the general formula is a valuable 'tool' in one's circuit solving 'toolkit'.