Well, 5 months after the last post, I just wanted to conclude with the solution, in case it helps another knucklehead like me. I pushed the car to the side of the house to store it for the winter and just got it our again as we had our first >70 degree day in Boston and the snow has *finally* all melted. I kept focusing on the starter because the accessories would work in the first position but when I clicked to turn it off it would all just die. I installed a starter relay (pretty easy to do - happy to provide tips if anyone is confused by what 30/85/86/87 mean). It still did the same thing.
I went to test the battery as I always have, but putting my voltmeter to the posts of the battery - it read normal/well-charged. Then, by accident, I touched the battery ground cable instead of the post and, sure, enough, the volts dropped to 7. So spidryell was right - it was corrosion (barely even visible) on the main battery ground cable. What happened was that it was enough to power the accessories, but when the starter engaged the whole thing just shut down. A little sandpaper cleanup and now it's all working again.
So the lesson:
- When testing the strength of your battery to diagnose electrical issues, measure at the cable end and not at the battery terminal. Thanks, all.
'82 Alfa Spider Veloce + '74 Fiat X1/9 + '10 Mazda RX-8 GT