In my view, unfortunately, the expert's answer is neither full, nor exact. Shewhart's cycle had 3 steps: Specification, Production, Inspection (see p. 45 in "Statistical Method from the Viewpoint of Quality Control". But it was just Shewhart who gave a sound explanation why the variability of all processes require using a cycle instead of straight line. So in Quality World Shewhart is the inventor of this cycle gist. Then Deming drew a four-step-cycle in his famous 1950 lecture, but as the experts outlined the letters PDCA were absent on that drawing. However, the essence of the corresponding actions presented there. So the Japanese did not invent PDCA cycle - they gave these names (letters) to the steps of the Deming's wheel. For those who are interested Henry Neave's book "The Deming Dimension" describes the history of creation and evolution of the PDCA cycle notion in detail (ch. 9).
The difference between "C" and "S" variants is the question of the interpretation of what's most important should be done. The letter "C" may be (not necessarily) interpreted closer for inspection while Deming insisted on studying the results.
Referring to the remote history of cyclic activity the priority should be given to Tao Hongjing's treatise (I read this translation from Chinese in Russian only: "The Art of Control" by Vladimir Maliavin).
The big article about the history and different interpretations of different versions of different cycles see in the Journal "Quality Management Methods", 2005, #3, 29-36 (in Russian)