Makes me wonder if the harness was designed for use with or without a horn relay?

Probably wouldn't matter much. The wire probably is a ground to the horn button. Or maybe hot to the horn. Either way, the relay contacts (normally open) would just interrupt the ground for the horn circuit, and the relay coil would get power from a hot source and be grounded on the other side would go the horn button. The relay just takes the load off the horn button contacts. Diagrams available on the web, but most seem to interrupt the power side of the horn circuit. That would be appropriate if the horn is grounded internally, but ours are typically two terminal - hot and ground, with the ground switched.