JB- Based on what you have said so far, there is no output from your alternator. I suspect after looking at your last pictures, that there is no input to the Field wire to the alternator. Next time you run the tractor, see if there is any voltage going to the "F" connector on the alternator, I suspect it will be zero. If so... that is where we need to start. I am guessing, based on mounting and wiring connector (ring lug), that what you have labeled as "Regulator?" is an aftermarket addon regulator. I suspect it is missing some wires and that is part of your problem. My guess, based on the wiring connector (6 pin molex) and the mounting, is that what you have labeled "What's this?" is the original regulator. It is still connected to part of the system, which is why you have the alt light on the dash, but no longer connected to the Field of the alternator and that is why there is no alternator output. If you notice, things that look original, are mounted to bumps stamped in the sheet metal, the suspected aftermarket item is bolted to flat sheet metal.
I hate to say say this... but at this point... I would trace out all the wires from the item you have labeled "what's this?" This would verify that it is indeed the 'old' regulator, we can see what goes where and what needs to be corrected. This is not going to be a fun task, but it is necessary before we put in a new regulator (which is what i suspect you need) and possibly blow it up,
lets try and avoid that.
After looking at all your posts again, it might be possible that your alternator is not the original one either... that is really going to muck up the troubleshooting.
It would be really good if some of the other 3054 owners would chime in and let us know what they have.