How to ensure an LED is completely off when pin is set LOW?
The LED should extinguish completely.  You must have a wiring error or are not driving it correctly in the software.

One possibility is that you have the pull-up enabled in the processor.

The AVR microprocessor has the ability to enable a high-value pull-up resistor (20k) if needed.  If you do this when you intend to turn the LED off it will glow dimly.

One way this can happen is to set the pin_mode() as INPUT_PULLUP.  It will also happen if you turn the pin to an input and write a 1 to the pin output register.

The usual way to connect an LED to an Arduino or other processor is to connect one end of the LED to the +5V with the other end connected to the output pin through a resistor in the range of 220-1000 ohm depending upon the LED and brightness required.

Some people call this resistor a current-limiting resistor, I prefer to call it a current defining resistor.  It is not there to protect the LED it is to set the desired current.

You write 0 to the pin to turn the LED on and a 1 to turn it off.