Split needs an explicit ApparentIntersections option like Trim has. Currently the use of apparent intersections by Split is controlled by the setting for OSnaps in Options > Rhino Options > Modeling Aids > Object snaps.
This is not documented in Help and generally opaque. It is confusing and can be very frustrating to the uninitiated. Rhinoceros Help
Why would anyone be expected to intuitively know that a setting for OSnaps controls how Split behaves?