To solve this problem, we need to determine the minimum number of moves Player $B$ needs to make to uniquely identify the triple $(x, y, z)$ chosen by Player $A$. The interaction between the players involves Player $B$ proposing a triple $(a, b, c)$ and Player $A$ responding with the distance formula given by:

$$D = |x+y-a-b| + |y+z-b-c| + |z+x-c-a|.$$

The strategy is to choose queries $(a, b, c)$ that reduce potential candidates for $(x, y, z)$ while eliminating ambiguity between solutions. Let us detail the moves:

### Move 1: Initial Query
Choose $(a, b, c) = (0, 0, 0)$.

The response $D$ simplifies to:

$$D_1 = |x+y| + |y+z| + |z+x|.$$

This response supports gathering information about the sum of pairs of the components of the true triple $(x, y, z)$.

### Move 2: Distinguishing Pairs
For the second move, choose a query that isolates one component more directly. Consider $(a, b, c) = (0, 0, 1)$.

The response is:

$$D_2 = |x+y| + |y+z-1| + |z+x-1|.$$

The change from the first to the second response helps compare differences and deduce possible values related to specific variables within $(x, y, z)$.

### Move 3: Final Clarification
Choose $(a, b, c) = (1, 0, 0)$.

The response is:

$$D_3 = |x+y-1| + |y+z| + |z+x-1|.$$

With these three strategic moves, it is ensured that the changes in response precisely pin down each possible combination of $(x, y, z)$, differentiating them effectively.

### Conclusion
Given the information from the three queries and their responses, Player $B$ can uniquely determine Player $A$'s triple $(x, y, z)$. Therefore, the minimum number of moves required for Player $B$ to be sure of determining $A$'s triple is:

$$\boxed{3}$$