I think this is a much simpler way of showing the same wiring. It's easier to understand.
You need 3 types of 12V power: un-switched (battery), hot with key on (called ignition in this diagram), and hot during start. It allows the pump to get power during start when pressure switch is not yet closed, then after start the pump continues to get power until pressure switch loses pressure.
Adding impact and/or roll over switches would add even more safety.