I see you got rid of the Arduino and transistor...then that leads me to ask, what did I need the transistor for originally?
The transistor is there because you don't want to run the current to power the lamp in that example through the Arduino. The light bulb needs more power than the Arduino pin can take without burning up. The small current from the Arduino is used to control the transistor which acts as an electric valve, allowing more power from the external power supply to flow through the light bulb.
Please note that the transistor only allows as much current as the external supply can provide to flow through the bulb, up to 100%. By removing the transistor you still allow 100% of the external power to flow with only the potentiometer acting as control.
External power flows through the pot to the light bulb to ground and there is your light circuit. When the pot is not turned up full, as that is wired, some of the power flows straight to ground which really you can disconnect that leg of the pot from ground and your circuit will still work and btw, waste less electricity.
Use a meter to check resistance between any two legs on a pot as you turn the dial. You need to use all three when you want to read the value on a pin but that doesn't mean it's the only way a pot can be used. With only two legs connected, a pot is just a variable resistor.
Do I have to test it? Do I have to check that 1 + 1 = 2?