It depended, basically. Most air forces required a witness to the "kill", who had to have seen the the pilot bail out or the aircraft crash - their wingman, or someone on the ground, or a gun-camera film. So in your scenario they would claim one "damaged" but wouldn't be able to claim a kill.

Exactly what the rules were and how stringently they were applied varied from situation to situation, and none of them prevented significant "over-claiming" during the war - either systematically for morale purposes, or just multiple reports of the same victory from different pilots.

This page from the US Air Force Historical Research Agency goes into a little more detail on the US side.