By inserting the USB connection to the mbed, I get a 4.65 V reading from the Vin and Gnd pin.

When I measure the wires connected to those pins from an external supply ( It is off ), the meter shows a reading of 303 mV.

I am guessing that there shouldn't be any voltage reading on the wires as the external supply is off. So what is causing that reading on the external cable ...

Leakage ? Remember your multimeter has an input resistance in the order of several Mega-ohms so it forms almost no load at all. Also a power supply which is switched off can hold a residual charge in its capacitors for several hours. At 303 mV most silicon transistors stop working so they can not regulate that residual voltage anymore. As long as there is no currentflow it is not something I would be worried about.

I have noticed the same effect. It showed up even with the external powersupply removed and it resulted in the LED lighting up that I use to indicate the presence of external power on Vin. I added an additional 1K load resistor on Vin and that seems to fix it.