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Consider the following expression:
Which of the following rewording expresses the above mentioned phrase correctly and in which contexts?
Note: The order in which I have written them is random and doesn't mean anything. If you feel one interpretation is preferred over the other, kindly mention so. If it is unchanged, you may want to mention that too.
Similarly for:
Each article is attached to the noun it immediately precedes. All of the phrases are grammatically correct; which one is actually correct depends on what you mean.
a page of a book - any page of any book.
a page of the book - any page of this specific book
the page of a book - this specific page of any book
the page of the book - this specific page of this specific book
Which one is meant by "the book's page.." or "a book's page.." is dependent on context. The article for "page" is not specified so it could be either. The context in which the sentence is constructed would determine whether a specific page or any page is being referred to.
Without any context I might assume that if you knew what specific book you were talking about you would also know what specific page you were talking about. So if you really want an answer, "the book's page.." sounds specific and may be more likely to mean "the page of the book", and "a book's page.." sounds general and may be more likely to mean "a page of a book".
As to whether you would say "a page from the book" or "the book's page" - again, this is dependent on the context of the situation and whether the page or the book was more important. Often we place the noun that is more important towards the beginning, although this can easily change depending on intonation. "The book's page" has fewer words and rolls off the tongue more easily, but really, it depends on the nuance you wish to convey.
A native speaker of English (UK or US) might refer to the book's cover, but never the book's page, because all books have multiple pages. We might say "A page of the book has fallen out" (the fact that it may get lost is, for the moment, more important than which page it is), or "Look at the page of the book with the list of names on it".
I can't tell you whether the book's page means a page of the book or the page of the book because, on its own, it doesn't really mean anything.
There are two excellent answers already, but they do not seem to have fully answered the OP's questions.
The book's page
is grammatical but ambiguous unless prior context makes clear which specific page is intended. In the absence of clues from prior context, a native speaker would be very unlikely to say it. It is not idiomatic in most circumstances.
It may mean
a specific but unidentified page in a specific and identified book
or
any typical page in a specific and identified book
Thus, the phrase is, in the absence of additional context, virtually meaningless and would not be idiomatic.
The book's fourth page is
makes clear which specific page is being discussed. Or
Most of he book's pages are
makes clear that we are talking about a typical page. Or
Some of the book's pages are
makes clear that we are making no claim about all of the pages.
All of the book's pages are
makes clear that every page is intended.
If you are asking whether
The book's page
is confusing, the answer is yes, which exactly why it is not idiomatic except in unusual circumstances. If you are asking whether English has the tools to disambiguate that phrase, the answer is "YES.
You seem to be looking for a difficulty that no native speaker over three years old will ever experience.
EDIT: The fact that this specific phrase is not idiomatic does not make it ungrammatical
The school's playground
is idiomatic on the presumption that the school in question has only one playground. Grammar deals with patterns. There are grammatical patterns in English (as I have pointed out above) to deal with a wide variety of situations involving the Saxon genitive. Being idiomatic means choosing the correct pattern for the situation.
A book's page has fallen on the ground., what will be your immediate interpretation, 1) or 2)? Note that there is no addressee in this sentence. It's just a general sentence not particularly addressed to any person.