Public use aircraft - pilot certificate  

Yes, that is correct, but the determination as to whether the OPERATION is public, or civil is a differrent matter and that is what determines whether an airman certificate is needed. A public entity can own an aircraft that meets the criteria for a civil aircraft and they can choose to operate it as a civil aircraft for one flight (airman certificates required) and also choose to operate it as a public aircraft on another flight (no airman certificates required)  As others stated, regardless of the strict legal requirments, operation of any type without the relevant airman certificates is foolish

On the other hand, it the aircraft did not meet the requirement for a civil aircraft, then all operations would have to be conducted as a public aircraft.