The distinction becomes clearer when all four forms are applied to the same definiendum, “human.”
Complete hadd: animal + rational — proximate genus + proximate differentia.
Incomplete hadd: body + rational, or rational alone — remote genus plus differentia, or differentia alone.
Complete rasm: animal + capable of laughter — proximate genus + property.
Incomplete rasm: body + capable of laughter, or property alone.
First ask whether the identifying feature is a differentia or a property. Then ask whether the proximate genus is present. These two questions classify the definition.