After lots of background processing and thought, I figured out why my part was not fitting on my Avanti for the backup lights. There’s a small metal threaded reducer that is situated between the backup light switch and the master cylinder. It's a larger diameter on one end and a smaller diameter on the other end. When I took off the old switch, this part was still threaded onto the old switch, and so the new switch didn't fit into the master cylinder correctly. Once I realized this, I unscrewed it from the old switch and then re-screwed into the new switch and now it fit perfectly and the lights work. Yay!
Here is a photo with the old switch still in place, in case it helps someone else who encounters this situation.