People had already given me the answer I was looking for, but as comments to the question. I'll just paste them here for posterity, and mark the question answered.
jlawler: 'Receiver' is not a grammatical term; I assume you mean 'modify'; and what you call a 'complement' is called an 'object'. Prepositional phrases can modify other phrases or whole clauses, as well as nouns. With an umbrella modifies the woman, a noun phrase; it's identificational, describing the woman. With a telescope modifies saw the moon, a verb phrase; it's instrumental, describing the means used. He saw the woman with a telescope is ambiguous between these two meanings.
Cerberus: Another useful term is scope, which more or less means "that which is modified", so the scope of the prepositional phrases is different between the two examples: in the first one, the scope of with an umbrella is (the) woman; in the second one, the scope of with a telescope is (John) saw the moon. Here is an interesting article about the scope of adverbial clauses, although it uses many technical terms at some point: Adverbial clauses, Functional Grammar, and the change from sentence grammar to discourse-text grammar.