These are max loads for given wire size, so the fuse protects the wire.
You would fuse according to the max maps that will be drawn from the circuit.
Yes, that was 14 gauge because it was carrying the load of the headlights, this is the problem.
Now it is only triggering the relay, a very small load.
18 gauge wire, 1or 2 amp fuse if you want one, really not needed.
Same for the ground side of the coil, 18 is fine.
A 10 gauge power to relay is a bit overkill, 12 would be plenty.
I would use the 12 to the headlights if you are running new wire to them.