However, I would say that real chess is sort of "2.5" D by this definition rather than full 3D, since the turn number coordinate of every piece can only go up, not down.
[1]: The game's formalism is slightly complicated by the fact that a piece could make a move of (x, y, t, L) -> (x, y, t - 1, L) which would count as moving one square in one dimension, but then the game's "system" automatically transforms that move into a (x, y, t - 1, L +/- N) move. (The choice of +/- depends on which player made the move. The value of N depends on how many other timelines there are at that t.) This happens because of the rule that the past cannot be changed, so altering the past requires branching the timeline.
Similar things also happen with a move of (x, y, t, L) -> (x, y, t, L2), because within a single turn there is a concept of whether the board has already had a move played on it or not. So if the board at L2 has already had a move played on it, this new move is instead transformed into (x, y, t, L3). So it counts as a move of (L2 - T) squares for the purpose of validity, but is actually a move of (L3 - T) squares.
I think you're right. Even if we classify classical Chess as 3D (2D board + a time axis for game states), time travel re-uses that same time dimension.
If you wanted to shift the complexity into ludicrous speed (this is inspired by archgoon's comment [0]), you could introduce another dimension by putting a classical chess board inside each square of the 4D board. The 4D move only succeeds if you win the game on the mini-board. Just, please, don't bring recursion into this, unless you're captain Kirk and absolutely need to nerd-snipe a rogue AI. (“They've gone to plaid!”)
There's another possibility I've hinted at by calling standard Chess ‘classical’. On second thought, let’s not go to Quantum Chess. 'Tis a silly place. (I tried to imagine what a Schrödinger's king would look like. I've seen things you people wouldn't believe. Queens checked en passant from the future. I watched knigths glitter in the parallel time-line near the Hadamard Gate. All those moments will be lost in time, like tears in rain. Time to resign.)
1. Horizontal within a board
2. Vertical within a board
3. Backwards and forwards along a timeline
4. Up and down across timelines
5. The timeline of how the game actually evolves, building out the tree. This is a bit of a stretch, since you can't move pieces through it in the same way as the other four.
For instance, if you have a stick you can push back and forth in a groove and count that as one dimension, if you push hard enough against an obstacle the stick might break and rotate out into the plane, but that's still only two degrees of freedom.
I suppose the actually intended analogy is that by riding a horse, the character is more evasive and can slip past another unit without being blocked. Also, I suppose that rooks must have wheels on, and a door that a king can only walk through in rare circumstances.