Do I have a faulty arduino uno, or have I found an Arduino bug?
If the Arduino "fails" the moment you apply power to a motor, you can bet that it is a power issue. 

Or possibly you are exceeding the output rating of the output pin. You should not draw more than 20 mA from each output pin continuously.


  I'm trying to decide whether I should invest in another arduino system.


Power the motor independently of the Arduino. This is normal advice for all but very small motors. Also make sure that you have a suitable driving transistor / MOSFET if the motor draws more than 20 mA. Take into account any surge startup current.



See Driving motors, lights, etc. from an Arduino output pin for some design ideas.

For example: