What is the best definition of key squares in king and pawn vs king endgame theory? Please provide graphics if possible. Is there a difference between "key squares" and "critical squares," or are these terms interchangeable?
The terms key squares and critical squares are sometimes used interchangeably in chess literature in king vs king and pawn endgames.
A square is called a key square when the occupation of it by the attacking king will secure a win regardless of which side is to move.
2nd through 4th Ranks
In this example, the key squares for the White pawn are e6, f6, and g6.
[FEN "8/8/8/8/5P2/8/8/8 w - - 0 1"]
A Pawn that Crosses the 5th Rank has More Key Squares
In this example, the pawn on f5 has two sets of key squares: e6, f6, and g6, as well as e7, f7, and g7. If White's king can occupy any of those 6 squares, they can secure the win.
[FEN "8/8/8/5P2/8/8/8/8 w - - 0 1"]
The Rooks Pawn are Exceptions
These positions show a draw regardless of which side has the move.
[FEN "5k1K/8/7P/8/8/8/8/8 w - - 0 1"]
[FEN "7k/8/7K/7P/8/8/8/8 w - - 0 1"]
In the first position, Black can simply shuffle his king between f7 and f8 and White makes no progress. In the second position, the move sequences 1. Kg6 Kg8 2. h6 K8 3. h7 stalemate. The sequence 1... Kg8 `. Kg6 Kh8 3. h6 Kg8 4. h7+ Kh8 with a stalemate, or loss of the pawn.
Endgame Simulations (against a computer)
I'm reading Fundamental Chess Endings by Karsten Muller and Frank Lamprecht right now, and the authors seem to imply there is a distinction between key squares and critical squares. On page 14, there is a list of symbols used in the text. A circle is used for a critical square and a star is used for a key square. So, there is an implied difference in this book. But I am puzzled as to what the difference is. "Key square" is defined on page 21 but "Critical square" is not defined before the authors start casually using the term. To add to the confusion, the authors use the circle symbol in diagrams on page 24 but refer to the squares marked with a circle as key squares.
Also, the terms key squares and critical squares are used in more endings than just king and pawn vs. king.
Various authors have used the terms key squares and critical squares differently. In some places Muller & Lamprecht's explanation is only fair. The confusion is that these terms have been used for two concepts. The first is the square the attacking king must occupy to queen his pawn in a K vs K-P endgame. This is well described above. The second concept is the square your king must occupy to capture a blocked pawn ( opponent's pawn blocked by your pawn ). These are the 3 squares on either side of the opponent's pawn. The Soviet Chess Primer uses terms key & critical squares interchangeably. It calls the 3 squares on either side of a blocked pawn the critical/key squares for capturing that pawn. ( page 146 , diagram 248 ). It explains all of this pp 141-150. Of course its a translation from the Russian. Muller & Lamprecht calls these critical squares. A column by Pandolfini in 2009 says the idea of key squares existed as early as 1860. Grigonev wrote about it in 1922. Strangely many early 20th century GMs did not explicitly express this concept, including Lasker, Rubinstein, Capablanca, Reshevsky and Fine. Alekhine apparently knew about it and Averbakh's "Pawn Endings" uses it.
I am reading Fundamental Chess Endings by Karsten Muller and Frank Lamprecht as well. As others mentioned, there is an implied difference between critical squares and key squares.
My two cents is that critical squares are relative to opponent pieces but key squares are relative to the game. Squares are critical squares to some of my opponent pieces (e.g. pawn) if she would lose those pieces were I to occupy the squares with my pieces (e.g. my king). Squares are key squares to the game if my opponent would lose the game were I to occupy those squares.