I see designated media area and DBT as the same. It is an area where if entered, bounced through, the ball is dead. If the player or any part of the player is on the ground in DBT ... Dead ball - 2 bases. Think about the inverse ... player in playable territory, ball in DBT. Player firmly grounded in playable territory but reaches across the boundary and catches a HR / Fould ball...... no one would question that the catch is an out. the opposite is then true.. player in DBT ball in play ... dead ball.
To anticipate the next question ... a ball in fair but player in foul ... the ball dictates the call because both are still in playable area.