The difference between running vs just ignition (with lights on too remember) is that the voltage is significantly higher with engine running;
higher voltage then means that the horn will draw higher current also.
The difference in why the PDM trips is probably from the surge current at initiation - the initial spike is probably just above the threshold in the detector circuit and trips that circuit.
It probably won't trip if triggering the circuit directly from the negative trigger - the way you have it is that the PDM60 circuit is actually already triggered, then you are suddenly going from zero load (relay off) to full surge load (relay on)
Again, just get rid of the relay altogether - it's completely superfluous with the PDM60
(ignore all the other circuits, which you should configure per your own requirements - but follow the wiring and programming for the Brown Circuit)
^ Note the programming for circuit 5 (Brown) is Ignition
AND Ground Trigger - i.e. it needs
BOTH to make the circuit active; so if you press horn button with ignition off, it won't come on; and the horn won't be on with ignition on until you actually operate the horn button.
You only need the Purple/Black wire from horn, the positive is not used