So here's the layout and test results today.
The house's main panel is mounted on the north wall outside. Two black wires supply power to the pump. Label them as L1 (lower) and L2 (upper). From the main panel, there's a conduit go to underground, that contains 5 wires: L1, L2, a blue wire, a white wire, and a green wire.
The pump, motor and timer switch are located on the south side of the house, roughly 100 ft away from the main panel. The underground conduit comes straight up to the timer switch box. Three wires comes from the conduit: 2 blacks (should be L1 and L2), and 1 green. Inside the timer switch, there're 4 places to connect, labelled as T1, T2, T3, and T4. The green wire connects to ground. The two black wires connect to T1 and T3 respectively. When the timer is turned ON, T1 is connected to T2, and T3 is connected to T4. T2 and T4 then power the motor.
Voltage Tests:
Flip the break on the main panel to ON, L1-L2 has 240V, but T1-T3 has only 120V. Then turn timer switch ON, T2-T4 also only got 120V to the motor. But the motor needs 230V.
Continuity Tests: with the help of two extension cords, I tested the following:
T1-L1 ~0 (good)
T3-L2 240K (open wire?)
My multimeter displays "0.L" when the two points being tested are not connected. Here T3-L2 didn't show 0.L but ~240 kilohms. I also tested the other two combinations: T1-L2 ~270K and T3-L1 ~160K. Not sure how to interpret these numbers. I thought they should be 0.L as well.
I also did the following, which is even more confusing:
L1-L2: if I put red pin of the multimeter on L1, black pin on L2, I got ~ 4 megaohms, but when I switched the pins, it measured a couple hundreds kilohms.
T2-T4: 0.L (good, as expected)
T1-T3: ~5 kilohms (big surprise to me. Is this a problem? if so, probably in the timer switch)
My questions are: (1) Does this mean that one of the wires, T3-L2, is open and I need to replace it? (2) If so, how difficult is the task? I'm totally new to this, want to fix it by myself which is probably faster and cheaper, but also don't want to mess it up in the process, (3) do other measurement numbers make sense? especially the 5kilohms between T1-T3.
Not sure if this kind of electrical questions belong to this forum or not. I just want to get my pump running sooner and guys here seem extremely knowledgeable and helpful. Let me know if I should take the questions somewhere else. Thanks!