Analyzing the Protocol
I would suggest to just bit bang the protocol. My interpretation of the shown pictures is:

Figure 1: This shows the protocol with a sync bit and a clock. A new bit is sent on every falling clock slope. (As you have figured out yourself)

Figure 2: The fast changing of the data lane might be just some noise. Maybe this is the end of the data sent from the printer to the cartridge

Figure 3: This could be the response from the cartridge. It takes a lot longer to send its bits after the clock. Probably its controller is a lot slower then the one in the printer. (This is only speculation for now, but it could make sense as the printer most likely requests data from the cartridge and not vise versa)

I would suggest to attach a micro controller to the lines and monitor the requests and answers that are sent. This would help to support the theory. And in the end maybe you can replicate it. Even without knowing the protocol.