When I look at the markings on your picture I see that a wire from "the starter" is connected directly to the horns. This wire is likely a battery HOT and would power on the horns as soon as the battery is connected, as you mention. This wire should probably be connected to one of the relay terminals.
The wire in your hand is labeled "to horn switch" on one end and "to ammeter" on the other end, which is also connected to the horns. This wire makes no sense to me.
The other wires (unmarked) that are connected to the horns seem to connect together and are not connected to anything. I believe these wires to be the horn activation wires that should also be connected to one of the terminals on the relay. This "Y" wire assembly should be the ONLY wires connected to the horns.
Your relay probably has 3 terminals. The 3rd terminal should be connected to the horn wire that goes into your steering column.
I don't have a diagram of how your relay terminals are laid out, but most horn relays with 3 terminals have a power feed (battery hot), an output (horn wires), and a ground (to the horn button). I found this generic diagram on the web, but it should be very similar to your setup.