I am not sure that what you are talking about is grammar. When I was in school, which was when dinosaurs still roamed the earth, we called it rhetoric, the art of being clear and persuasive.
A basic rule of English is that modifiers of nouns or pronouns should be as close to the noun modified as possible.
A second rule is that adverbs modifying a verb should precede the verb but follow the subject or come after the the object of the verb.
I quickly crossed the street
I crossed the street quickly.
Another basic rule of English is that participles (though formed from a verb) may act as part of a verb phrase, an adjective, or a noun. What role is the "dangling participle" playing.
And a fourth rule is that certain words can be omitted from a subordinate clause if the words are so obvious that no meaning is lost.
So, rhetorically, if you mean that the woman was crossing the street when she was hit by the bus, normal rules of interpreting English suggest that
While crossing the street, the bus hit her
means that the bus was going from one side of the street to the other, an act rare but possible for a bus, but contrary to the intended meaning. If the participle is adjectival and applies to the woman hit, it should be close to "her" rather than "bus."
The bus hit her while crossing the street
is not, however, best analyzed as a participial phrase modifying "her." It is better analyzed as an abbreviated adverbial clause indicating the time of the action indicated by the verb. And adverbs frequently come at the end of a clause.
I think your second version should be intepreted as
The bus hit her while [she was] crossing the street.
Notice that there is no problem with
While she was crossing the street, a bus hit her.
So, although I agree with you that "while crossing the street" is much better at the end of the sentence than at the start, the reason is not that it is an adjectival phrase, but rather that it is an adverbial clause.
EDIT: The comments have indicated a lot of confusion about terms.
A “participle” is an inflected form of a verb that may be used as part of a verb phrase, as an adjective, or, in some cases, as a noun.
A “phrase contains no subject and, except for verb phrases, no complete verb.
A “participial phrase” is a phrase that contains part of a verb, namely just the participle.
A “clause” contains a subject and a complete verb and any necessary complements.
A phrase can act as an adjective. In which case, it should be as close to the noun being modified as possible.
Subordinate clauses can come at the start or end of a sentence.
Some participial phrases are best interpreted functionally as abbreviated clauses. This is especially true if a participial phrase is headed by a conjunction.
A “dangling modifier” is a participial phrase occurring at the start of the main clause that, according to the normal rules for placement of modifiers of nouns, applies to the subject, but is meant to apply to something else.
While crossing the street, the bus hit the old woman
That, according to the normal rules of construction implies that the bus was crossing the street. But what was undoubtedly intended is
The bus hit the old woman while crossing the street
So the “dangling modifier” here is a participial phrase that is misplaced. I suppose a dangling modifier could occur anywhere, but I agree with your references (and Ronald Sole) that dangling modifiers (almost?) always occur at the beginning of clauses and so get mistakenly applied to the subject.
Why does this happen so often?
I think it is because participial phrases are frequently, in terms of function, truncated clauses, and clauses have their own subjects.
While the old woman was crossing the street, the bus hit her.
No confusion there about whether the bus or the woman was crossing the street.