I don't understand your question. Gaining the diagonal opposition is precisely what white did in move 6-8. That's what you wanted, right?
To get the diagonal opposition white has to make a triangle with his king. He can go either Kf4-e4-f3-f4 or Kf4-f3-e4-f4 or Kf4-g3-f3-f4. Therefore 6. Ke4 and 6. Kf3 and 6. Kg3 are all correct.
This is puzzle number 77, page 88 from the book practical chess endings. I'm not getting the logic of this game. On move 6 the book recommends white to play king k4 but my question is why doesn't white gain the diagonal opposition? The black king is at rook 3.