You will need a wiring diagram to diagnose the circuit. Some models used a relay, most just powered the valve with the starter signal. The starter signal also heated a bimetalic switch till it openned; shutting down the circuit on the ground side. The time it took to open the switch was determined by the engine temp. At temps above 59deg F the switch was open and there was no circuit and NO coldstart above 59deg.
Here in Florida we never see a cold start for months at a time!
Bosch Master, ASE Master, L1
33 years MB technician