Learn English – She said to the Headmistress,”Madam, forgive me the fault and let me appear at the exam”

Imagine that when person "A" says "this" or "that", he and the person spoken to (person "B") are looking at the same thing. It's as if they are pointing at the thing with a finger.

Now, when someone else (person "C") is reporting to person "D" the conversation between A and B, person C cannot point at the thing with his finger because the thing is not there, but is with A and B. So person "C" cannot use "this" when reporting speech between A and B.

A: I like this kitten.
B: -- I like that kitten too. Here, kitty kitty! Ouch! This kitten has sharp claws.

C: "A" said he liked the kitten.
D: --Did "B" like the kitten too? What color was it?

Here, the "thing" is not something with fur but an intangible:

A: You would be wise to remember this: a stitch in time saves nine.
B: -- I'll remember that.

C: A told B that he would be wise to remember: a stitch in time saves nine.
D: -- Did B say he would remember it?
C: I forget what he said.

The problem you see here is caused by your having outgrown the “rules” you were taught.

Teachers commonly treat “indirect” speech as a representation of the original speaker's actual words, because that makes it much easier to explain how the “backshift” of verbforms works.

In some cases sentences of the form X says/said Y stick as closely as possible to the original speaker's actual words, but there is no requirement to do so. What the subordinate clause Y in such a sentence expresses is the content of the original speech, not necessarily its form. For instance, I may quite legitimately write:

Hamlet asks himself whether it is more honorable to endure ill-treatment or to resist it.

In fact, Hamlet asks nothing at all like this: he asks

Whether ’tis nobler in the mind to suffer
The slings and arrows of outrageous fortune
Or to take arms against a sea of troubles
And by opposing end them.

So you cannot recreate the original speech from the indirect or reported version. Likewise, there is no “rule” governing how you report a speech like “Will you please help me?”or “Will you lend me your books?”—you have many choices open to you. And there is no conflict within your sentence pairs—they report the same content in different ways.