Two duff motors Lexi. Those caps (capacitors) normally exist in two states. Good or exploded to bits
They do fail on the Chinese compressors fairly readily but they let you know normally with a kaboom! and spilt guts. I've replaced a few in the past.
That fact the capacitor works OK on another shows it is good. Also a known good cap does the same thing.
I'd say the motor has a weakness and they all display the same fault.
You're sure it's wired correctly?
My diagnosis would be that the main running winding is open-circuit and that the reason it runs slowly for a few seconds is that it's only running on the start winding. Basically, the capacitor is effectively a short-circuit when power is first applied and thus it can feed a high current into the start winding to get the rotor initially spinning. As the capacitor slowly charges its resistance (actually reactance) builds until it is effectively an open-circuit and thus reduces to almost nil the current in the start winding. By this time the rotor is tyurning fast enough that the main running winding can maintain rotation and spin it up to working RPM. The running winding on its own cannot get the rotor moving from rest, hence the need of the capacitor and start winding.
The value of the start capacitor is not that critical. It's voltage rating and internal protection is the most vital part. If they went short circuit the motor would burn out very quickly as the start winding is not indended to be energised for more than a few seconds.
Old fashioned motors that did not use capacitors used a centrifugal contact set to switch out the start winding. If the running winding on these went open circuit, the speed never got fast enough to switch out the start winding and they burned out.
I recall a true tale of an old boy who used an under-rated motor on a big sawbench. He thought he had the most wonderful motor. He told of how it would change gear as he was sawing a particularly tough piece of timber and the motor slowed. What was happening was the motor was slowing enough due to the heavly load to switch the start winding back into operation, hence the motor seemingly changing gear...
needless to say, that motor did not enjoy a long life...
Lexi, I hope that sheds some light...