Learn English – Changing reported speech into direct speech

If the situation reported is still valid at the time of reporting, backshifting is optional. See Swan (2005) Practical English Usage, pages 275-6, for several examples.

John said that he would give me a pen. Correct in all situations.

John said that he will give me a pen. Possible if John has not yet given the pen to the speaker.

S2: I wanted to let you know that he had sent you an email. Correct in all situations.

S1: I wanted to let you know that he has sent you an email. Possible if the sending was so recent that 'you' may not yet be aware of this.

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.