***** NOT A TEACHER *****
In my opinion, learners who ask about the parts of speech want to pass their examinations.
So I agree that students should follow the models in post #4.
On the other hand, people who happen to be interested in analysis should consider the comments in post #5.
I have never taken a university-level linguistics class, but I have heard from time to time that the whole preposition vs. adverb matter is not so simple as the usual secondary grammar book rules indicate.
Here is just one comment from one scholar:
"So close ... is the correspondence [between adverbs and prepositions] that many grammarians, beginning with Aristotle, have refused to recognize them as two different parts of speech. Prepositions are described as adverbs that take objects, and the preposition is viewed as no more different from the adverb than the transitive verb is from the intransitive."
He goes on to say that "There is logic in this approach, but [he admits that] the fact remains that the term preposition [is] very convenient."
He also gives this example: "He shot at the lion."
a. Is "at the lion" a prepositional phrase that modifies "shot"?
b. Is "shot at" a transitive verb with "lion" as its object?
i. After all, he points out, we can say "The lion was shot at."
Authority: Paul Roberts, Understanding Grammar (1954), page 229.
Here is what another scholar opines.
1. He came across the bridge.
2. He came across.
The scholar says that -- in the view of some scholars -- "there is no reason in principle [my emphasis] why [#2] should not simply be classified as [a preposition] which can occur with or without [my emphasis] a complement."
Authority: Rodney Huddleston, Introduction to the Grammar of English (1984), pages 348 - 349.
*****
In my opinion, learners should definitely follow the traditional rules regarding the difference between prepositions and adverbs. The goal is to pass the tests!
On the other hand, I believe that all of us should let advanced learners know that the whole area of parsing in English is not so simple as the traditional textbook would have us think.
I have found that when it comes to analyzing English, we should do so with a huge dose of humility.