You might have dealt the most damage to put him into final stand, but you failed to deal the finishing blow. You are only rewarded for dealing the finishing blow even if the other person put him into final stand, the person that failed to finish him off should only get a assist not counted as a kill because you did not kill him.

They should fix final stand to where they are not immune to damage during when they fall down. If they fix it to where they can take damage during the fall it would enable you to kill the said person. That's why i currently call final stand in mw3 final pain killer because its like final stand on top of pain killer from mw2 combined.