The paper does focus on photosynthesis, however, because the Nocera's lab works on "artificial photosynthesis," a field largely devoted to understanding the electrochemical processes of photosynthesis and mimicking them in man-made systems. For the most part, "artificial photosynthesis" focuses not on carbon fixation but on the transformation of light to electrochemical energy (via electron/proton transfer). This is the process that Nocera's research is replicating, and likely why he focuses on using solar cells as the source of energy, to parallel a plant's use of solar energy as a route to energy storage/O2 production.
Finally, I think some of your scientific analysis needs correction:
Electrolysis (in its simplest form) doesn't require any solute - it can be performed, very slowly, in pure water. Adding salt (say, NaCl), as you mention, speeds up the rate greatly. However, in doing so, the reaction changes - you'll evolve hydrogen as before, but at the anode you'll evolve chlorine gas, not oxygen. You can't use electrolysis alone to produce oxygen gas or as a route to storing energy. One alternative to this is to use an electrocatalyst, but currently, (as the article mentions), the only available catalysts require expensive metals or very basic environments (or both). This article is on the discovery of a new catalyst that will allow for the production of oxygen via catalyzed electrolysis at neutral pH and with a cheap, abundant catalyst (cobalt).
Also, if we're looking at the same quote: "storing energy for use when the sun doesn't shine," the article is talking about humans storing energy, not plants. Even so, plants do store energy for when the sun doesn't shine. They metabolize hydrocarbons into ATP via mitochondria, just like we do - so glucose and other products of photosynthesis are indeed forms of stored energy.
I think a lot of the confusion stems from the fact that we're reading a popular article and not the actual research - these articles tend to pick up on terms like artificial photosynthesis, energy production, and solar energy and run with them.