von Mises: cylinder, circle, or ellipse?

The answer is…all of the above!   The von Mises cylinder is centered about the {1,1,1} direction in principal stress space.  The ellipse that we learn about as undergrads applies only to plane stress (where the third principal stress is zero), and this is just the intersection of the cylinder with the plane.  The circle applies to the octahedral plane, which is the view of the cylinder down its {1,1,1} axis.  This animation should clarify what is going on:

MisesCylinderAndMisesEllipse