You need to differentiate between what will work, and what is correct, and decide which you will do.
You should check if the current pump is operating at 230 Volts or 115 Volts. If it is 230 volts, then the wires to the switch should have 230 Volts between them.
A pump will work without a ground. You shouldn't try to operate it without a ground.
If you use a 2-wire pump, you could use the third wire as a ground. You should identify it as a ground wire. Two-wire pumps are more common than they used to be.
With a 30 year old pump and wire, I would replace the wire as well and use the correct wire.