The "difference that one passes along no matter what, while the other will be handled by exactly one object" is not what determines whether Structural or Behavioral.
Structural patterns ( as per Gof)
are concerned with how clases and objects are composed to form larger
structures. Structural patterns use inheritance to compose interfaces
or implementations.
Behavioral patterns ( as per Gof)
are concerned with algorithms and the assignment of responsibilities
between objects.
It does seem the distinction is academic in the case of Decorator vs Chain but the fact remains that Decorator complies with the definition of Structural.
Chain also uses linked lists but its not essential the list items are related by inheritance but the list is about assigning or taking responsibility.
That's the reply as per GoF, but I don't believe its worth the time for new student of patterns to waste too much thought on the distinction in this case, especially given the similarity of the end products!