A mushtarak is a word with more than one separately assigned meaning. The printed argument assumes, for analysis, that “al-shams” may be assigned to the solar body, its rays, and their aggregate. This is a hypothetical test of the definitions, not a new dictionary prescription for ordinary usage.
If the entire body-and-rays aggregate is intended, the word indicates the whole through correspondence. Within that whole, the rays are a component. But under a separate assignment to the rays themselves, the same word would again indicate an entire assigned meaning by correspondence.
The question is therefore not only “Which word?” and “Which signified item?” but also “Under which assigned meaning and in which respect is it being signified?” Ignoring that qualification creates the apparent overlap.