Number of possible heaps on $\{1,...,2^h-1\}$
The definition you give looks like the definition of a complete tree. With the restriction that nodes are in $[\![1, 2^h-1]\!]$, then it is also a perfect tree of height $h$.

Instead of looking at the leaves, you should look at the root of the tree: since the tree is perfect, it has two children of size $2^{h-1}-1$ (this is how $C_{h-1}$ appears). You just have to think of how to split $2^{h}-2$ keys between left and right (the key of the root is uniquely determined, since it is a heap).

I will edit this answer if it is not clear enough.