What is the purpose of the electrical device at the interior refrigerant lines?
Liquid line solenoid valve. When valve closes, pressure builds up in line and a high pressure switch at outdoor unit turns off compressor and refrigerant flow.
You had the name correct but not its function.
Some times we refer to the liquid line solenoid valve as a pump down valve both names mean the same.
The thermostat operates the solenoid valve not the compressor contactor when these are in place. The valve is normally closed with no power from the thermostat or when the thermostat satisfies. Thus creating low pressure at the A/C condensing unit where a low pressure switch is installed that is wired in series to the compressor contactor. The low pressure switch actually becomes the operating control for the compressor.
Basic operation for this type system thermostat opens the solenoid valve increases pressure to the outside unit the low pressure switch closes on pressure increase and starts the compressor.
Your statement about the pressure building up when the solenoid valve closes is out of tune with reality. In Reality when the valve does close the head pressure actually decreases because it is located in the liquid line. If this solenoid valve was installed in the high pressure vapor side of the unit and it closed, the pressure would definitely increase and you would blow out the discharge valves within the compressor and if installed a high pressure cut out switch would shut the unit down.
Keep in mind a high pressure switch is never ever used as a operating control only as a safety switch and usually has a manual reset button.
pumpdown solenoid.
Must be a high-end system.
This older version may not be a “pump down”.
Depending on the location of the condenser, these valves were used to stop migration of liquid refrigerant during the off cycle. They just shut when the compressor shuts down.
It prevents overloading of the compressor at start up.
This requirement was later handled by a Thermostatic Expansion Valve that did not have an internal equalizer.