I can see errors that will cause problems and I need to study the diagram and the write-up to give you a correct answer free of errors but it may take a bit of time so I'll have to get back to you.
Aaaahh...ok.... yes I can see what you mean about confusion there.
Normal operation of say the high pressure switch would be as follows.
Saying minimum and maximum is where I think the confusion starts.
Take the high pressure switch for example.
Normal description of operation and adjustment...
I think you need to get the operation straight first, as the explanation of operation of the circuit and the operation of the pressure switches (minimum and maximum adjustments)does not match what you are telling us now.
The hp switch only activates to shut the system down.
Not according to this .........??
When minimum pressure is achieved on the high pressure side, the high pressure switch will activate.
Contact racing might be a problem. For example ,as the HP switch activates K1, K2 looses it's hold-in power. (that's as far as I went) Sorry, should have read a bit further BUT the contactor on LP will de-energise and re-energise which is not good.
I think you need to hunt for a model number or at least a photo of what the internal connections are as it might be you have wrong connections.(frame connection on the bottom left for example)
I realise some don't agree with the sonic approach but you might want to take a peek at this device...maybe it will suffice, maybe not.
https://www.sparkfun.com/products/639
As far as I can make out, these units are designed to run off a "driver" with varying level of input voltage from the mains. I don't see why you need to run them individually from a transformer (which it appears you've misquoted the specs) To run off solar would then be simple to run from an...
Suggest you look here.... https://www.arduino.cc/en/Tutorial/ArduinoToBreadboard
No mention of the crystal or the caps in your description so I'm guessing you didn't include them.
Don't see any cathode band.....I'd be inclined to think it is a small tantalum capacitor especially if it has a + sign on one end.....bit difficult to see any markings really. Perhaps a better pic?