Here is a diagram that should be more intuitive. The solid lines should be +12V and the broken lines should be 0V (ground).
Relay No.1 is only controlled by the 'Heater Control Unit,' which is the A/C and fan control switch on the dash. Relay 1 is only powered when the Ignition switch is on, it cannot run after the car is shut off.
Relays No.2 and No.4 are peforming the exact same function. Think of them as one relay. Connecting the dashed green line to ground will turn the relays on, which will give power to both fans. There are a few different things that can enable these relays, including the Data Link Connector, ECU pin 3D, and the Coolant Fan Control Module on post-recall cars. Relays 2 and 4 are always powered by the battery, they can activate the car even when the key is not in the ignition.
Relay No.3 switches a second ground to both fans. This will change the speed from low>> medium or from med>> high. Relay 3 is always powered by the battery, it can increase the fan speed even when the key is not in the ignition. The line that I marked with dotted pink is going to be ground when the solenoid is on. When you measure it, you should be seeing 0V on your multimeter when the relay is switched on. You might never see 12V on this wire during normal operation
Here's a table showing how the relays work together to control the fan speed:
To turn a relay on, ground the signal wire (bottom-left on each relay in the diagram). For instance, to turn on Relay No.1, connect the thin Violet/Pink wire to ground. I used a piece of stereo wire, connected to the negative battery terminal. You'll hear a click when the relay turns off or on. Relays No.2 & No.4 are activated with a thin Green/Black wire, and relay No.3 turns on with the thin Blue/Green wire. Remember that most of the relays can activate the fans even if the key is not in the ignition.
-s-