Standard Tic Tac Toe is -- Impartial or Partisan?

I am currently studying basic game theory (combinatorial) and was introduced to impartial games.
The "definition" of impartial games I saw was: "An impartial game is a two-player game in which players take turns to make moves, and where the moves available from a given position don’t depend on whose turn it is." (Also, the distinction between players is done by who had the first move)

Now consider a game of Nim, given a state, the next possible states would be same irrespective of who's turn it is.
In a standard game of chess, given a state, the next possible states would depend on whether its "first player's" turn or "second player's" turn. So its not impartial, i.e. partisan.
In a standard game of tic tac toe, given a state, the player whose turn it is, is fixed by who started first. So how does this definition apply here? (One way might be to say that the set of moves of one player is empty and the other set is the standard one, hence partisan).

But I was taught that the game of Tic Tac Toe is impartial, which I can't get my head over. So what is it that I am missing?

Edit: As mentioned in a comment, if we assume that the rules don't require, for a given state, the next possible moves to be restricted for a particular player (X/O), still, the next possible moves would be different for both the players, right? Since player 1 can only use Xs and player 2 can only use Os. (So it should still be Partisan?)

Solution 1:

You can always turn a partisan game into an impartial game, by adding a status which is called "current moving piece".

E.g. for tic-tac-toe, you can put a coin beside the board. Each move consists of putting an X or an O on the board, and then flipping the coin. If the coin is head-up, then only X can be placed; otherwise only O can be placed.

Under this formulation, the game rule makes no difference for the first player or the second player, thus it becomes an impartial game (*). You can see that this game is equivalent to the original game, in the sense that the coin records which symbol can be placed by the next player.

_{(*)There is a further complication that the game can end in a draw. But let us pretend that there is no draw game.}

Thus every partisan game can be "impartialized". The impartialized game has, as all impartial games, a nimber which is nonzero if and only if the game is a win for the first player.

However, what makes this pointless is that this procedure leads to weird behavior under "sum" of two games. Therefore it is almost impossible to study a partisan game via its impartialized version.

Recall that if $A, B$ are two impartial games, then their sum is a game $C$ which has one copy of $A$ and one copy of $B$, and a legal move in the game $C$ is to make either a move in $A$ or a move in $B$.

This notion is used in an essential way in the study of impartial games, c.f. Sprague-Grundy theorem.

Now if we consider the sum of two "impartialized" tic-tac-toe, then it becomes a weird game. Whenever the first player makes a move on board $A$, the optimal strategy of the second player is simply to make the exact same move on board $B$.

That is, the second player no longer needs to respond to the move in the original game. The coin now has no way of recording who is going to make the next move on that board.