The confusion is because it appeared that the circuit and the LED next to it appeared to come from the same article. The circuit was wrong and did not match the type of LED.
Here is a circuit that I think will do what you require.
View attachment 94415
The value of the fixed resistor and potentiometer will have to be calculated using the data sheet for the LDR. The potentiometer 's value should be about the same as the dark resistance of the LDR. The fixed resistor should limit the current to less than the LDR's current rating. You need to calculate the value of "R" to match the current rating of the LED and it's wattage to match the power dissipated in the resistor. You may have to adjust the value of the resistor slightly from the calculated value to get exactly the current you want through the LED. The left hand side of the circuit does the switching. It shuts down the current regulator when the LDR is illuminated. The right hand side is a current crude regulator to control the LED current. I have not built this but it should work.
Les.