> Either we have to say that the position does not dictate the possible moves, or that this does not fully capture the position.

It does not fully capture the history needed for determining future claims of draw by repetition. But by definition, the position fully captures the position.

The notion of position used by the FEN notation [1] includes the board diagram, side to move, castling rights, en-passant options, as well as the number of halfmoves since the last capture or pawn advance, and the total number of moves. The last one or last two are often ignored in everyday notions of position.

[1] https://en.wikipedia.org/wiki/Forsyth%E2%80%93Edwards_Notati...