Difficulty in updating the balance factor of nodes in AVL tree
The reason that the number of possible values for the balance factors at $y$ and $z$ is limited is not only the fact that the tree has been AVL before insertion, but also the precise location where the rotation is performed in an ex-AVL tree.

Rebalancing is only needed at the lowest position where the balance becomes $\pm 2$ where below had balance factors $0$. All those $0$ balance factors have been changed into $\pm 1$ depending on the side where the key has been entered. The short reason for this is that when a node has originally balance factor $0$ and one of its subtrees becomes taller after insertion, not only the balance factor becomes $\pm 1$ but also the total length of the tree at that node becomes one taller, so we have to consider its father. 


If that father has balance $0$ it will become $\pm 1$, the tree is taller, and we proceed    up again [diagram b)]. 
If the father has balance $\pm 1$ we may happen to improve the
balance to $0$, and we stop, no rotation needed  [diagram a)]. 
If the balance becomes $\pm 2$ we rebalance at that point using the
proper rotation. After rotation the tree at that point has the same
height as originally, so we do not have to consider balance factors higher up
in the tree.  [diagram c)]




So the issue here is to determine the context where the rotation takes place. This is determined by following the insertion process bottom-up, from the leaf where the key was added. 

The precise type of rotation (single/double, left/right) is determined by the insertion path that was followed below the node where we rotate.  

In the single rotation case we have left-left insertion from $x$ so $y=-1$ (longer to the left, my balance factors are opposite to yours). The value at $z$ is not important, its complete subtree is copied during the single rotation at $x$.
In the double rotation case we have left-right insertion from $x$ so again $y=-1$. At $z$ we might have any value: $\pm 1$ depending on whether the key went left/right at $z$, but even $0$ in the special case $z$ itself was inserted!