The rules for kick-offs are different from the rules of punting.
1) Kick-offs occur to start the game, to start the second half, and also occurs after a score.
2) A punt is what happens on 4th down.
When the ball is kicked off, the receiving team must
field the ball unless the ball goes out of bound or out of the back of the end zone. If the receiving team does not field the ball, the kicking team can pick up the ball and take possession. Your example would be a turn over by the receiving team and an instant touch down by the kicking team.
When punting the ball, the rules are different. The receiving team does not have to field the ball. The receiving team can choose to allow the ball to go into the end zone and choose not to field it or catch it. On punts only, that would be a touch back.
The receiving team would get the ball on their own 20 yard line. I believe that 20 yard line rule was changed a couple of years ago to the 30 yard line.
Kick-offs have to be fielded by receiving team while the ball is in the field of play or the kicking team can pick the ball up resulting in a turnover
Punts do not have to be fielded by the receiving team. The ball will be spotted where it stopped or where the kicking team touches the ball. This does not result in a turnover. The receiving team gets the ball.
The term "punt" and the term "kick-off" are totally different terms with totally different meanings in football.