Please help with exhaust

[06YarisRS]

I had some exhaust work done when the header was replaced. I have tried to recreate what the exhaust system looks like now as compared to what I think was there originally. As you can see, a long cylindrical section (part B in my diagram) was eliminated and another part ( Part A, resonator?) moved further rearward. I am not sure what these parts are. My exhaust is going in for a leak and a section of the rearward pipe is being replaced. I have a current code P0420 and a Toyota dealer hs said that it can occur with an exhaust leak anywhere in the system, even much farther downstream. I had had a P0137 O2 sensor circuit low voltage and I replaced the sensor. I now have fluctuating voltages and have read that it is supposed to be steady voltage for the downstream sensor. The P0137 is gone but now I have the P0420 code Catalyst system efficiency below threshold

I am not really sure how to proceed. I think I should replace the pipe section I got and have and be sure there are no other leaks. I can do this by having someone hold their hand over tailpipe and lsiten for leaks. I found one using this method. If my P0420 code does not go away, I planned to use the anti-fouler spark plug trick to eliminate the P0420 code. Air/fuel ratios are good, so I don't think it is having any adverse effect on my engine. Fuel economy is great as is power.

My question is that at least one part (cylinder shaped - resonator?) was removed and I'm wondering if this is causing my engine light (P0420) code. What is the name of the long cylindrical part that was removed from my exhaust system in my picture?

If I get my new pipe section installed and make sure all the leaks are eliminated, am I on the right track? I think that replacing the entire front pipe would be the proper way to go, but that would be super expensive.

Can you please tell me definitively what parts A and B are in my labeled image?