Did this system work in the past with 4 post solenoid and them fail, or was 4 terminal solenoid added after problems developed.
There are 2 different configurations for the Ford style 4 terminal solenoid. The difference is in the configuration for 2 small posts. The posts may be labeled S and I
Type 1: Solenoid has activating voltage input to S terminal and other end of internal activating coil is grounded through the mount. “I” terminal receives 12 volt battery voltage when solenoid is activated and sends this 12 volts to the ignition coil to bypass the coil resistor and give a hotter starting spark. Only used on gas or propane engines, not diesel. The “I” terminal has nothing to do with activating the solenoid.
Type 2: Voltage is applied to S terminal and I goes to ground. Either or both can go through a switch.
Neutral safety switch should be open when gear shift is not in neutral, and closed when shift is in neutral. One side receives voltage when start switch is activated to start position and current should flow through NS switch when gear shift is in neutral. This should activate the S terminal of your first solenoid and turn it on to supply voltage to the switch on the starter.