No, that doesn't really cover corresponding squares.
Corresponding Squares and Silman's Endgame Course
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DVRazor is right (almost).
2.Kf3 ! is the correct move instead of 2.Kg3 ?, but Black should respond 2...Kg7 (but neither 2...Kf6 as indicated in the text, nor 2...Kg6 as suggested by DVRazor because of 3.Kg4 +-) to 2.Kg3 to draw the game thanks to the distant opposition.
Moreover, after 2.Kg3 ? Kf6 ? 3.Kf4
f4 is not a "critical square" per se; instead in this position, e6-f6-g6 are genuine critical squares, ie squares allowing White King to win the Black Pawn on d6 whoever is to move. Of course on the other side, a6-b6-c6 are also critical squares.
3...Ke7 4.Ke4 ? is again a mistake
An outflanking move on the g file was necessary to save the win : 4.Kg5 ! was the best answer.
After 4.Ke4 ? Black could save the game by staying on the e file : 4...Ke8 !
Actually the point for Black is to be able to respond ...Kf7 when White play Kf5, thus threatening at the same time to go to one of the 3 critical squares mentioned above. Kf5/Kf7 is a reciprocal zugzwang meaning that f5/f7 are corresponding squares, and so are g5/g7.
So basically the opposition is important on the f file (and on the g file) and not really on the e file where White has no access to the e5 square. On the Queen side the situation is quite symmetrical with a5/a7 and b5/b7 being corresponding squares, meaning that opposition is important on the a and b files.
It is also easy to see that to be in time with the White King on both sides, the Black King should stay on the same file as the White King's one when on the c-d-e files. As said before, on those files the opposition is useless. That's why after 4.Ke4 ? Black can draw by 4…Ke8, but also by 4…Ke7. This means that e4 and e8 are not corresponding squares despite the opposition.
Consequently in the main line after 4.Ke4 ?
4...Kd7 ? is wrong (4…Ke8 ! is a draw)
5.Kd3 ? is wrong (5.Kf5 ! is a win)
5…Kc7 ? is wrong (5…Kd8 ! is a draw)
6.Kc3 ? is wrong despite Opposition (6.Ke4 ! is a win)
6…Kb6 ? is wrong (6…Kc8 ! is a draw)
7.Kb4 ! Yes, at last this move is the right one (opposition on the b file)
You made also a couple of important mistakes in the line 1.Kg2 Kf6 :
2.Kg3 ? is wrong (2.Kf2 ! is a win)
2…Kf5 ? is wrong (2…Kg5 ! is a draw)
and finally the move 10.Ka6 is not a "Triangulation", it is just an outflanking move.
Conclusion : the Opposition has been considered for more than one century as a particular case of corresponding squares and I'm afraid YOU NEED TO KNOW THIS CONCEPT.
The concept of Opposition DOES NOT COVER Corresponding squares but the concept of Corresponding squares DOES COVER Opposition.
This endgame study (by Dedrle) illustrates perfectly the fact that Opposition has to be used carefully and that it is NOT VERY easy to understand when and how to apply Opposition rules and maneuvers, especially if you omit the theory of critical and corresponding squares.
Finally, if you just trust Opposition, why not playing 1.Ke1 with opposition ?
Instead, those who are aware of the theory of critical and corresponding squares understand that 1.Ke1 Ke8 ! is a draw whereas 1.Kg2 ! is the clue, aiming at invading the critical square g6 and taking the opposition on the f file as soon as the Black King goes on it.
I'm creating this thread because all of the other threads about Corresponding Squares are at least 6 months old and dead.
I suck at the Endgame and so I decided to work on it a little. I watched IM Daniel Rensch's video series on the King and Pawn Endings. All went well until the topic of Corresponding Squares which I did not understand at all. The Wiki article in Wikipedia went over my head as well. I had looked a couple of times in Jeremy Silman's Endgame Course but the term Corresponding Squares isn't in the book at all. Nor is the term in the glossary on his website. So I read his sections on Opposition, Beyond Basic Opposition, Triangulation, and Outflanking. And I found out why the term isn't in the book or on his website.
YOU DON'T NEED IT!
He explains the concepts very easily in his Endgame Course.
Direct Opposition: The kings are 1 square away from each other either on a file, rank, or diagonal. Whoever moves when this occurs DOES NOT have the opposition.
Distant Opposition: When the Kings are an odd number of squares away from each other on a file, rank or diagonal, whoever moves DOES NOT have the opposition.
Opposition without Direct Connection: When the Kings on the same file, rank, or diagonal, draw box around the two Kings. Each king will be an opposite corner. All of the corners will be the same color. Whoever moves DOES NOT have the opposition.
Triangulation: One King moves to the side and then steps forward on their next move, forming a triangle. This is to give the opponent the move and YOU the opposition.
Outflanking: A Maneuver where the King moves to the side, placing a file between the Kings (sometimes giving the enemy King the Opposition.) This allows the Outflanking King to move forward towards his goal without allowing the enemy King to take Direct Opposition.
So to sum up.
Direction Oppositon: Kings 1 square away, enemy to move.
Distant Opposition: Kings an odd number of squares away on same file, rank, or diagonal, enemy to move.
Opposition without Direct Connection: Put both kings on same color, enemy to move.
Note: Direct, Distant, and W/O Direct Connection ALL require the kings to be on the same color square, enemy to move.
Triangulation: Step your king to the side, to give the opponent the turn, and step forward forming a small triangle.
Outflanking: Put a file or rank between you and your opponent's King.
That's it. VERY easy to understand.
These concepts working together covers Corresponding Squares as far as I can tell. Here's several examples. I hope I did it right.