The Uno page says this:
The board can operate on an external supply of 6 to 20 volts. If supplied with less than 7V, however, the 5V pin may supply less than five volts and the board may be unstable.
Looks like you need to put 6 or 7 in, not 5, but I'm not sure if that's true for going in via Vin or not. Apart from that afaik that's a legitimate way of providing the power to the board and the motor.
You picture isn't what you explained in the first post, which showed the PWM going direct to the low side of the motor and I didn't see that working. The transistor approach, a la my link, is good.