Use of "of" to define objects?

I argue that the "United States of America" is different than the "Kingdom of Cambodia", for instance, because one uses "of" to describe a relationship with something larger (ie: the continents of America) while the other is describing an object that is one in the same (ie: the Kingdom IS Cambodia).

What kind of terminology would apply to these differences?

Solution 1:

• a picture of Bill
  = a picture that depicts Bill
• a picture of Bill's
  The word of comes as close as it is possible to get to having no meaning whatsoever.

It's the default preposition, used in thousands of idioms
(like the complex quantifiers few of, a lot of, and oodles of, for instance)
and often mistaken for the genitive case because of its frequency.

In fact, it has lots more grammatical uses.
For instance, when nominalizing the verb in a clause,

• The passenger in compartment 3A departed early.
• The porter identified the passenger in compartment 3B.

the verb's most prominent argument (normally the subject, for intransitive clauses,
and the direct object, for transitive clauses) gets marked with of
(while a transitive subject gets a genitive -'s suffix):

• the early departure of the passenger in compartment 3A
• the identification of the passenger in compartment 3B
• the porter's identification of the passenger in compartment 3B

That's one use, at least as common as the possessive marker.
In picture nouns, to give another example, various possessives have different uses:

• Bill's picture
  = some picture that Bill owns; or that depicts him; or that is associated with him, somehow.
• a picture of Bill's (ditto)
• a picture of Bill (= a picture that depicts Bill)

So any argument based on the literal meaning of of
is unlikely to convince anyone, since there isn't any.