How can I add lighting using 14/3 wire to an existing 20A circuit?
You can not use 14 awg wire on a 20 amp protected circuit it would violate code and create a fire hazard.
One example is 14 awg wire can fit into back stabs and even on 15 amp circuits these regularly fail on a 20 amp circuit they would be a fire hazard.
You can use an extension cord and there are several reasons this can be safe. Extension cords are temporary and not made for multiple loads like the premises wiring is.
I bet you did not know that 14 awg extension cords usually have a higher ampacity than the wire in your walls. NEC table 400.5.A.1  Example 14 awg has an 18 amp current rating for most types and some are even 20 amps. Why the conductors are not enclosed by the structure and don’t have insulation around them.
So although it may sound ok it is a huge code violation and liability if there was a fire and someone got hurt you would be the one responsible.