Those pictures, as labeled, are misleading to a degree.  Although the information is true, what really causes (most of) the distortion is distance, not focal length.  To fill up a frame with a face you need to be at different distances for different focal lengths, and that - the perspective distortion caused by being close, or apparent foreshortening caused by being far away, is what you are seeing in those shots.

There is also some degree of distortion inherent in the lens itself, and that will vary not only with focal length but with the particular make and model of lens.  No general statement can be made about it that is universally applicable.

To answer your question as directly as I can, with an APS sensor you have to stand further from the model to fill the frame with the face than you would with a full frame camera.  Consequently the predominant distortion effect you see will be caused by that change in distance.