This is an entry to the Monthly Topic Challenge #3: Pencil and Paper Games
The last line contains a critical clue.
Curiously, Hints 1 and 2 contain a critical clue.
Start by making sense of the "Given" section. You can disregard colors, the bottom section, and anything to do with "tic-tac-toe" or "ciphers".
The wordplay does not give the mechanism, but rather the genre.
X to play. The boards were selected somewhat arbitrarily; the possibilities are endless. Your task is to determine what makes them special.
What very well known game has the symmetry found in row 2?
Added a clue to the image. Why might a teacher use a red pen?
Stiv has correctly deduced that the first 9 boards are "valid" according to certain rules, but the next 9 are invalid.
Let's focus on row 3. Stiv has correctly guessed that it is not a pawn. In fact, the board is already solved for you (it is an example solution-imagine if it first looked like a row 4 board).
Can you solve row 4? Hint: a board's validity is independent of adjacent boards; the initial layout was only as a hint to the name of the game.
Each invalid board can be solved with a brief visual inspection; having a red and green pen in hand simply clues the mechanism. Nothing to do with checking; this puzzle is about valid boards. 3 of the 8 boards in row 4 are pawns.