Erro in Chess Mentor pawn endgame course

I'm following the course: Pawn Endings: Beginner to Expert, and the lesson: double pawns

This is the position, at move 5, I played Kc4, which is commented in the following way: "5.Kc4 looks natural, but after 5...Kd7 6. Kd4 Ke6 7. Ke4 Kf6, White can make no progress and would have to head back to b4 and start over again!" But Houdini gives a checkmate in 21 moves from Kc4, and the real mistake made by the FM and IM who wrote the course is the move 6.Kd4, which actually draws.

Here I put the chessboard and the lines given by Houdini, because this is an endgame worth to be learned.

And in 0.1 seconds the tablebase gives 1.Kd4 as a win in 23.

http://www.k4it.de/index.php?topic=egtb&lang=en

Seems like they didn't check these positions.  I wonder if they got the analysis out of an old book or something.

However, I found more easy to learn the 1.Kd4 line, than the one they were teaching. Because I tried them against Crafty, and the one they were teaching was impossible to win, I would always run into an opposition I didn't see. While the 1.Kd4, it is impossible to miss the win.