Can you post your entire circuit.
The statement that it works and "However when I remove the connection to the computer the whole thing goes crazy printing lots of pulse times which are all wrong as I am not pouring water, and thus should not receive anything." seem mutually exclusive.
One wire will not work; there is a sneaky earth coming in somewhere.
If your bucket tips quite slowly (say, as slow as 10ms which is 1/100 second), you have plenty of time to poll
for a change, double or triple check the signal is stable to allow for debounce, increment your counter, then continue to poll until the end of the pulse. An arduino runs its main loop about 20000 times a second. Polling allows debounce,
interrupts make that quite a bit harder. Try a quick polling sketch to see.