I think it all comes down to what the writer/speaker thinks about the reader/listener

ie. If it is new to the reader use 'a', otherwise use 'the'

as for your examples

So I just cannot see the difference between "this is a book I told you about yesterday" and "a woman who fell 10 meters from High Peak was lifted to safety by a helicopter."

Why can a definite article be used with "woman" for introducing for the first time and not with "book"?

If you told me about the book yesterday, then it wouldnt be new information so it would be: "this is the book i told you about yesterday", Using 'a' in this case almost implies that I forgot about it, and you are reminding me

For the second one, I have never heard of the[1] woman (in the article) before so you describe her as 'a' woman and not 'the' woman.

the[1] last examples:

It should be "the" according to the explanation people have given me, right? Unless there is more than one letter he sent to Burnham on June 20, which is unlikely.

This does read to me as if there is more than one or that he wrote so many letters over the years that they can only specify which one they are talking about by date.

Likewise, it should be "the" unless Burnham wrote several letters that contained a description of the meeting.

it is 'a' because he wrote only one letter. "Burnham wrote one letter to Olmsted...", if you said 'the' in this case, it would imply that I knew about the letter before you said the sentence

[1] I use 'the' because you know what I'm refering to here