Hi Stjepan,
First, so that everyone understands what you are trying to describe, you need to refer to your "vents" as fans, and the "jack" as a plug. In the case of what you are describing, it is actually a free socket, but we still call it a plug. English is not a logical language!!! I think your English is very good.
We do have ventilators, which can be fans, but to we English, the vent in this context generally means the slots where the air comes out to blow into the car or up to the windscreen. We also have jack plugs, but these are generally for audio units or low voltage power supplies.
A jack is what we lift our cars up with.
Now to your problem.
The item you are asking about is the high pressure switch for the air conditioning refrigerant compressor.
It is a two stage pressure switch. As the pressure rises in the high pressure output from the compressor, if the pressure rises too high because the refrigerant is getting too hot, because maybe the cooling for the condenser is not enough with the fans on half speed, it switches them to high speed. (They will already be running half speed if the air con is switched on)
If then, the pressure continues to rise, even after the fans are on full speed, then the next stage switch interrupts the feed to the magnetic clutch, and switches off the compressor.
So to answer your question, yes, if this pressure switch was faulty, it could affect the operation of the condenser fans, as could anything wrong in the plug or wiring connecting to it. Also, if it has gone completely wrong inside it, and has jammed in the fully operated position for both switches, then the compressor would not run because it would wrongly detect too high a pressure in the system and switch off the clutch.
The fact that your previous meddling around with the plug into this unit fixed the problem for a while points to something wrong in this area.
To change this switch, you will lose any gas in the system, and need to have evac and gas re charge. Do not take out the switch untill you have the replacement to fit immediately. Do not leave the system open to air.
For information, there is also a separate low pressure switch in the system to detect loss of refrigerant.
HTH
Roger