I don't think a horn relay is different than direct hookup to battery for test purposes. A relay is a switch with big contacts and is operated by grounding a wire at the wheel. It keeps the major current switching way from horn button.
That type horn has a vibrating metal diaphragm. Electromagnetic device. Flexible contact......on off. Creates sound magnified thru the "horn". As said , clean contacts, adjust the sound screw, get shinny good ground to body of horn. Make sure the horn bracket has shinny ground to horn body. Use whatever fuse is called for. If it tests good with direct to battery but not with relay/horn button, those components are faulty as Gail says. Works with a 20A and burns a 30A after install? Would expect the other way around. Bad relay? All in all, they are not expensive to replace with a good donor.
Is this a 12V converted truck? Is this a 12V horn? 6V horn will work, just curious. Without a cover, it's all rusty, corroded, gummy...
There is no horn relay shown on 1951 wiring diagram. your relay link to J. Carter says "Fits just below the voltage regulator on left side of firewall. 6 Volt only. Began use in 1953 when the horn changed from the bell design to the ram horn style". You are showing a ram horn, horn. Your horn is not an original for 1951. No big deal to me and probably you, but FYI. All wires on all trucks should be fused or fusible linked, except the big batt cables. Why not? get rid of ammeters too, they are fire starters. Switch to Volt meter.

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