The EFI main is the relay that will send power to the injectors, idle control and o2 sensor heater. The circuit opening relay usually was bolted to the ECU in most cases
The big relay called main is the car body system main relay - sending power wherever is required, it is a 5 pin relay and grounds out all circuits when the relay is switched off. - Please note the grounds out part, it does not mean open circuit.
the 3 fan relays are used to make the fan run in 2 speeds, these relay switch them to parallel and in series, in parallel they run at full speed, in series they run at half speed. The toyota fan control relay input is always loss to ground (backwards relay) meaning it remains ON when you switch on the key - the fan does not run with relay on, When the engine warms up the relay is switched off and the fans run.
The reason they dont run with car off is because the relay called "main" switches off.