For my horn button, with the new harness, I soldered the center contact to the disc (time will tell if that turns into a mistake). As stated, there should be spring pressure pushing the disc up. Also I've noticed the disc, when pushed up into that rubber cover that I have, it stays put - like shown in 19555-5's image (note: again, my center contact is soldered to the disc). Then when you put the rubber cover onto the wheel nut, there should be a gap between the disc and nut. Both (spring under the wire/tight fit inside the rubber cover) should be contributing to pushing the disc out away from the nut. As described, when you push down on the rubber cover, the disc makes contact with the nut.
Replacement parts may not always be the same. Vendors will use different suppliers whenever needed. When I look at Walck's replacement cover and disc (with nut) I see what appears to be a dimple in the middle of the disk. If you have this, I'd assume the raised part should be pointed down so the spring can push it up and keep the disk off the nut. Just the spring alone may or may not be enough keep the disc up but if the cap doesn't hold on to the disc, an option could be to try a bit of glue, e.g. rubber cement, contact spray, etc. to get the disc to stick, or solder a few burrs on the edge of the disc to get a bite into the rubber cover. However, looking at Walck's full horn button kit, the disc appears flat. Adjusting and tweaking: Definitely!