In the context of choosing a jump towards the center, I thought the diagonal move (kosumi) was also called the diagonal jump, although that's a muddy term. Might not the elephant move be explained as "large diagonal jump" ? Yi-lun Yang/Phil Straus use "diagonal move" + "double diagonal."
Who cares. Trying to give a name to each shape fails necessarily sooner or later. It is an indirect connection, regardless of the shape.