To me. the key to running around your backhand to hit a forehand (my image of an inside out forehand) is the footwork:
1) turn sideways, as if you are going to hit a forehand (even though the ball is still well to your backhand side) , and then
2) back pedal **explosively** to line up the forehand with the ball
You will see this footwork in many of the above videos in this thread.
This footwork pattern allows you to get set and move forward, during the stroke, into the ball for greater power,