The other test you can do (beyond testing continuity through the wires of the old clock spring) is to test the voltage at different places through the horn wire path. For that you would need to hook up the battery and use a different setting (DC voltage) on the multimeter. The black probe goes...
https://www.fluke.com/en-us/learn/blog/digital-multimeters/how-to-test-for-continuity
Basically you want to see less than 1 ohm resistance between two ends of a wire. The clock spring is simply like a bunch of wires that can rotate around when the steering wheel turns and still be connected...
As shown by the link I posted, part number: 8430624010 supersedes the other part numbers you posted. Supersession(s): 84306-24010; 8430620040; 8430628020; 8430630060; 8430633020; 8430650080; 843102L080. And it fits your LX 450. I'm not saying that you need to buy it from Lexus, but pretty sure...
I think you are mixing up earlier and later models. What you have in your picture is a clock spring, but the plunger part you bought is for a '94 and earlier. '95 and above use the clock spring, no plunger...