For this concept you are trying hard to solve a problem, when in fact you should try to remove the issue at hand. You have too much IO. I think that multiplexing is very creative, but unnecessary. What if the jars had 2 positions instead of 4? What if there were less jars? Have you looked into machine vision for sensing?
You did a good job going through the different possibilities, and the advantages and downsides of each. I am not an electrical expert so I cannot really comment on the specifics of the design, but I do think your goal of reducing the number of microcontrollers and to simplify the wiring as much as possible is the way to go. You could also make it so that not all jars rotate, some jars can be static, and only one or two jars per row rotate in order to match the image, which would greatly simplify the electronics as well.
I dont understand circuits very well, but your descriptions and graphics did a fantastic job of explaining them. I love how you started from a simple introduction into the jar circuit and worked your way up through iterations beyond that.
-Good call on moving from the simple circuit. Design for fab would be very difficult if each jar had to have its own micro controller and circuit board.
-What testing have you done so far on the matrix circuit? It seems well thought out, and there are some predicate devices (keyboard), but have you done anything to show that it will work for your application?
-On a side note--there are some typos/weirdly phrased sentences in your descriptions. Its no big deal, just keep that in mind for your next website.