Once upon a time, Bob was telling Alice about a chess match: "One turn before the end only 4 pieces were on the board, and White could guarantee checkmate for themselves in several ways". Once Alice has learned the number of ways, and the names (and colors) of these 4 pieces, she was able to uniquely determine position of each figure on the board. I bet you are even more clever than Alice and don't need to know all these little things. What are these pieces and their positions Bob was talking about?
1. That was a usual match with all chess rules applied. Thereby 2 of those pieces must be Kings.
2. All the information Bob gave to Alice is mentioned in the puzzle formulation.
3. Alice knew only name and color of each piece, no other information about them; like starting position of a pawn or a cell color of a bishop - nothing of it.