You can check the continuity of the pin to the horn relay without the ring. If the continuity is not there, you know that something between the pins is not making contact. You can also check the ring to the horn relay pin with the ring in the installed position. Check the contact of the spring plunger contact to the slip ring as well. Start with everything assembled, and work you way to where you have continuity. The last part you took off will be the missing connection, and either that part or the one it touches is the culprit.
If you can disassemble the pin from the turn signal switch, you can verify the shape of the Dorman part vs. the original one.
I have some issues with my horns, and did not consider that the pin you are looking at would be the culprit. I will have to check mine.