The identified components are:
1. Fan, if it has two wires the only way to provide a slow speed is by dropping the voltage from 12 to 6 (?). If it has three wires, they are individualy energized to provide the two speeds via differnt windings in the fan motor. Wire colors are (?)
2. Temperature switch, three wires (green, blue white)
3. Relay, two wires (probably a temperature activated switch as it is on the hottest part of the radiator)
The pictures have changed my `overnight' thoughts.
I
believe the `relay; also has an internal resistor (single speed two wire fan only). 12v from the battery goes to a fuse then to the relay, then to the fan, then to ground to complete the circuit. The resistor is in the circuit to drop the voltage to ~6v so you get 'slow' fan speed. When the temperature rises sufficiently, the `relay' will bypass the the resistor and allow the full 12v to run the fan at `high' speed.
The three wire temperature switch MAY prevent the fan from operating until the water temperature at the bottom of the radiator reaches 85-90° C. The bottom of the radiator is the coolest place and that is where the coolant returns to the water pump. Without tracing the wires I can not offer any other help.
I would determine the color of the wires, follow them if possible and search for the fuse that feeds the circuit. Then try and draw a schematic, noting wire colors and connections.
As the fan is not dependent on the ignition switch, using your multimeter with an aligator/crocodile clip on the black lead (-) fastened to a good ground and a common straight pin or safety pin on the red lead probe the appropriate wires to determine when various parts of the circuit are enregized and if there is in fact 6 or 12 volts present. Pushing the pin into the insulation will not cause any significant harm to the wires.
It is also possible that the ignition switch permits initial fan operation but one of the connections to the temperature switch will bypass the ignition switch to maintain power until the temperature drops to a safe level when the key removed from the ignition.
When checking across the terminals of components that are `switches', a "Zero" voltage indicates a `closed' switch, 6 or 12 volts indicated the switch is open. The voltage is truly named "Potential" as it CAN do work.
I wish that I was there `looking over your shoulder' but that is obviously impossible. My `instincts' point to the `resistor' (wherever it is located) as the cause of your other electrical trouble as well.
