«هُوَ الرَّجُلُ الْكَرِيمُ»
He is the generous man.
«هُوَ السَّائِرُ رَاكِبًا»
He is the one traveling while mounted.
When the definite predicate is accompanied by an adjective or circumstantial qualifier, any restriction must be interpreted with respect to the qualified description. «The one traveling while mounted» does not claim exclusive possession of every possible kind of movement.
An additional attribute, state or adverbial phrase narrows the predicate. If a restrictive meaning is intended, it applies to that compound meaning rather than all possible applications of its head word.