"Sam broke." (Before you say this makes no sense, consider: "After six hours of interrogation, Sam broke, and confessed to the crime.") "Sam broke into the house." 'Broke' here has a completely different meaning than in the first sentence--yet 'broke' is still an intransitive verb. Calling the prepositional phrase here a modifier seems inadequate--it doesn't modify the verb, it changes it completely. There are lots of examples of context determining the sense of verbs (and other parts of speech). "Sam ran." "Sam's nose ran." "Sam broke down." (In his car? Or did he cry? Two completely different meanings.) Is there a term for how context determines the meaning of verbs, in different usages? And where a preposition is involved, how do we distinguish this from a phrasal verb?
Pragmatics is the technical term for
The branch of linguistics dealing with language in use and the contexts in which it is used, including such matters as deixis, taking turns in conversation, text organization, presupposition, and implicature. —Oxford Dictionaries Online
In Real Life (as opposed to textbooks) utterances do not ordinarily appear outside a specific discourse, and that discourse furnishes a context which constrains the interpretive possibilities a hearer brings to bear on every local part of the discourse, not just the verbs.
For instance, the name Sam in isolation is even more polyvalent and ambiguous than the verb broke—it might refer to Sam Gamgee or Sam Weller or Sam Jaffe or the Sam who played catcher on my son's middle-school baseball team or any number of Sams known and unknown to the hearer. But within a specific discourse it will be presupposed by the hearer to refer to a specific individual who (we may assume) was previously identified in the discourse. That is an application of pragmatics.
Similarly, the prior mention of interrogation provides a context which narrows possible interpretations of the word broke and makes it easy to infer the meaning "cease to resist". That, again, is an application of pragmatics.
Situations like your example where the context may initially be different than what the reader is expecting, and require a second reading/different interpretation to understand, are referred to as garden path sentences.
A garden path sentence, such as "The old man the boat," is a grammatically correct sentence that starts in such a way that a reader's most likely interpretation will be incorrect; the reader is lured into a parse that turns out to be a dead end or unintended.
In the example above, reading "the old man the boat" is confusing to a reader, since by reading 'the old man', we imagine an elderly person who will be described/is the focal point of the sentence. However, the closing part of the sentence doesn't fit. The actual intent of the sentence is to indicate that the boat has a crew of elderly people, i.e. using the word 'man' to mean 'staffs/crews a vessel'. You should avoid these constructions wherever possible.
Verbs like 'broke' and 'run' are ambiguous. It takes context to disambiguate them.
Many philosophers and linguists would say that it is the phonological forms /run/ and /broke/ that are ambiguous and that there are actually two words for each form ('run1' and 'run2', 'broke1' and 'broke2'). Context helps listeners figure out which word is being used.
Contrast this with words like 'I' and 'you', which are not ambiguous, but are context-sensitive (or idexical).
Context can be used for disambiguation (as with 'broke'), or as a hard-coded part of an expression's meaning (as with indexicals).