In your video, the band is definitely turning the correct direction. You can see the weld or other light spot passing by, indicating the direction- which in your case is indeed towards the pivot. (Which is how virtually every horizontal bandsaw I've ever seen does it.
And yes, it's very easy to get a blade "inside out". They're rolled for shipping- that is, twisted into three loops- and it's easy to try and "undo" that loop and wind up with the blade inside out.