What's the difference between "neural" and "neuronal"?

I think the distinction you provide is correct, although I would not talk about neuronal stem cell.

A NSC (neural stem cell) can produce 3 things: another NSC, a neuronal progenitor or a glial progenitor.

The progenitors then develop to neurons or glia (astrocytes or oligodendrocytes).

Another example is the neural tube, the structure of the embryo that will develop to form the nervous system. Again here you use neural as a wider term.

Interestingly, in biology you talk both of [biological] neural and neuronal networks (although it seems to me that neuronal network is more used), but in informatics you talk about [artificial] neural networks.

Neural is the adjective for nerve and neuronal is the adjective for neuron.

Neural means dealing with nerves or the nervous system. Neuronal means dealing specifically with neurons.