Sounds like 2 problems:
Service level detection not working properly
Pilot not working
Both of those systems have a ground in common. We would recommend checking the ground connection to the OpenEVSE controller board and the AC Input grounds.
The chips that do the detection of L1 or L2 are connected to AC_Test lines. If you reset the station while it is plugged into the vehicle the Auto detect is disabled. The relays are only cycled if the vehicle is not plugged in.
That does not explain the problem of the station changing from Connected to charging.
There are only 3 things that could affect the detection of state from Ready, to Connected to Charging.
For proximity I would verify when the handle is plugged in you can hear the switch click. If it does not click the system may think the button is still pressed down.
For proximity the key is hearing the click when you RELEASE the button after connecting to the vehicle. If the button remains in the closed state the vehicle will never start charging.
The behavior you describe is correct.
If OpenEVSE detects a vehicle is connected (via the pilot line), OpenEVSE will skip cycling the relays to auto detect the service level and default to Level 1. As soon as the vehicle commands the relays close, then the AC_Test lines read voltage on the AC_Detect chips and the service level is set to the correct value.
Based on your descriptions the pilot is fine. I would still check the button is opening (Clicking) when you release the button after inserting into the vehicle. You may also want to check the connectors and termination on the relay, if the vehicle does not see voltage or the drop is too high it will reject the session and change from "Charging" back to "Connected".