Yes that basically proves it's at the column IF the wire from the horn contact area to the relay is viable. That later.
So to recap the assembly.
1. First goes the spring loaded contact plunger down in it's hole.
2. Next is the disc. As you know it is dished. It is a spring. A large belleville washer. The OD of the dics touches the plastic part of the steering wheel. The ID will be up. As in: the middle is raised around the nut.
3. The three legged plastic part's legs go thru the cup, thru the dics and against the steering wheel hub, held by the three screws. Be careful with this part, it is easy to get crooked or not thru all pieces. Don't over tighten the screws.
The assembly has to accomplish a sort of floating cup held up by the disc. You can test it without installing the horn button. Push down on the cup. This makes the disc touch the metal around the nut at the same time it is in contact with the plunger. The plunger touches the "column bearing/slip ring" which has a wire down the column and out to the relay. You are grounding the relay terminal as you did in the test but this time by "touching the wire" to the column metal, which is grounded, using all these parts. Be sure the metal around the nut where the disc will touch is clean and clear of rust and dirt. Make sure the disc itself is clean and shinny and still has a cupped, flexible, dished shape.(not flattened out with age). If it honks, put the button in the cup.
You don't need to disconnect the battery if you don't want to. You are only dealing with ground. If horn sticks on, be ready to pull off the ground wire at relay. Gently.
You don't need to pull the steering wheel off for this.........yet
In your list of parts, you did not mention the Plunger
