How are you testing, with a multi-meter or a non contact tester?
A non contact tester (the kind that lights up when held to the wire) is not reliable for testing, it will not tell you voltage and can show phantom voltage on wires with no power. You must use a multi-meter so you can properly test voltage. Your pump may be wired for 240v, you absolutely cannot tell whether you have 240v or just 2 hot wires without actually measuring the voltage. A 240 volt motor trying to operate on 120 will run, usually at less than full speed, sometimes it's hard to tell. It will get hot and trip the thermal protection. Looks like your motor can be wired to use either voltage, but requires changes depending on what voltage is provided.
In a 240v setup it is possible for your timer to interrupt only one hot wire, It is acceptable to start and stop a 240v motor that way. This means power is going to the motor even when the timer is off, so you must be very careful when testing this setup. Testing each lead separately will not work, you must test across the leads to confirm you have power from both phases and actually have 240v.
A 240v motor only needs two hot wires providing 240v to operate, it does not need a neutral but the timer or other controls may operate at 120v which could require the neutral. For safety a 240v motor should always be bonded with the green ground wire, but will still run without having that connected.