Why do we need double-rotations to rebalance AVL trees?
I think you are right for this example. But the title of your question is misleading. Let me explain.

After insertion the tree can be rebalanced with one (single or double) rotation at the lowest node $x$ where unbalance occurs. There are four cases, as you mention. If the key has been inserted in a left-left or a right-right subtree we do a single rotation to the right or to the left, and if it was the left-right or right-left subtree we do a double rotation at $x$ (in suitable direction).
In general a single rotation will not work.

Back to the example you show us. The lowest point of unbalance is $T$. From $T$ the new key $W$ was added right-right. That means a single rotation left at $T$. As far as I see the result of this is exactly what you have drawn in your diagram. 

To close, your title is misleading, as we do not need double rotation here, but we need one in other situations.