The outcome is the same if White King is on d6, so we will see only this scenario. The pawn advancement gives the opposition, however the King located on the 6th rank changes the final result of the position.1.e5...1.Ke6While this move naturally wins with the idea of Kd7 or Kf7 after Black moves, there's also another idea.Kf82.e52.Kd7 would also be winning.Ke83.Kf6Kf84.e6Ke85.e71....Kf8While Black takes the opposition now, White has a pawn move available!1...Kd82.Kf7and White controls the promotion square.2.e6Ke83.e7Kd74.Kf7And White promotes winning the game, so in conditions when the King is on the 6th rank, and the pawn not yet, there it's winning either way.