The crown molding lengths that are running parallel with the vault are going to be in a different plane than the lengths running perpendicular across the vault if you install the molding so that it is resting on the wall and ceiling the way it was designed to lay.
Consider the fascia boards going up the rake of a roof and how they are clipped off at the bottom because the piece is traveling in a sloped plane and the miter has to turn back to a level plane, so it is a compound cut and the miter is longer than the miter of the board on the level plane.
It's the same principle with the crown, and the transition piece on an inside corner would give the look of clipping off the bottom of the crown molding going down the slope back to a level plane. Hint: the transition piece will have an inside corner cut to miter with the level crown and an outside cut, cut right side up on the saw, to miter with the piece running up the vault in a sloped plane.
In the image posted, the pieces that are running across the vault are not laying properly against the face of the wall; especially the one on the high side of the vault would be visibly off the wall. JMHO, but when I see crown installed like this I think it looks tacky.
You'll need to use transition pieces to turn the crown from one plane to the next if it is going to be installed properly. Those pieces can look silly also, especially on an outside corner.
That being said, if it is small enough, you can fudge the angles and the way it is designed to lay to make it work with some sanding and shaping of the miters.
I've heard of using differing widths of crown with similar profiles to make these miters, but have never actually tried that one.
You could also use corner blocks for both the inside and outside corners if that is acceptable.
I realize this isn't a tutorial like you were looking for, but hopefully it helps you visualize and understand better whatever tutorials you come across.