Are you sure all the wires are in the correct place? You've checked to ensure you are getting 240v to the contactor and beyond? With 240v you should get 120v between each hot wire and ground and 240v between the two hot wires.

Here is how I tested the contactor. Turned on the unit from the thermostat to energize the coil so the contactor closed.

I got 24v on the contactor coil voltage. Got 240v on the bottom hot wires incoming power from the disconnect box. Got 240v on the other end of the contactor (coming out). Got 0v from each bar of the contactor so that means they are closed circuit. Got 120v between each hot wire (in/out) and ground.

I got a different reading if I tested without turning on the unit from the thermostat.

One question though. What is the correct way to test the contactor with or without the unit on?

Don't assume the capacitor is good because it is a year old. New capacitors have been known to fail prematurely.