That can't possibly work. The L293NE does not have a 5V output. It requires 5V input for the internal logic. Perhaps you misunderstand the wiring.
Indeed I did - thank you for the clarification 
So, if I'm understanding correctly, the 12V 1A power supply (via the power port) is powering the Arduino directly, the H bridge, as well as the motor via the VIn.
Do not attempt to power the motor from the Arduino 5V output pin, as it can damage the Arduino or interfere with its function.
Would you recommend I power the Arduino independently of the motor (not via the VIn)? Does this cause harm to the Arduino or lead to interference?
Thanks