When did depictions of Buddha shift from the relaxed, calm figure to the joyful fat guy?
The "joyful fat" Buddha is actually not Gautama Buddha (the original Buddha and the founder of Buddhism), but rather, is an entirely different man - Budai. Budai is a Chinese folkloric figure who came to be identified with Maitreya. Maitreya is not a historical individual, but rather, is an incarnation of Buddha who is prophesied to appear on Earth sometime in the future.
I don't know enough about East Asian Buddhism to really answer why or when Budai became identified with Maitreya, but I think it is worth noting that the "fat Buddha" is not the same person as the calm, relaxed Gautama Buddha. Budai is popular in Japan (where he is known as Hotei) as well as in China, yet there are quite a few statues of Gautama Buddha in Japan as well.