Aaron it's good to see you posting here.
I hope that you have already taken the dive into the previous conversations here on the early cartridge types.
Lancaster is a great source for documents and images from Europe. Take a look through his posting history and you'll find a deep trove of treasure on this topic.

The use of the extractor tool is simple.
Into the small hole at the end of the tool is set the firing pin of the cartridge.
The little balls seen in the drawing (I've seen variations of shape there) are used as fulcrum, one or the other bears again the breech face of the barrel and the fired case is pried out of the chamber by the pin.
The hook could be used to reach in and grab the case by it's mouth in the event that the case head separated from the case body.

The drawings and images of them that I have seen appear to gage the ring size to the case head size. I can only imagine that the ring end could be used to re size the thin metal case head with the pin removed.