We can use the future perfect to say that something will be completed or achieved by a certain time.

time here is a count noun: meaning a moment in time or a point in time.

See ODO, definition 2.1, and the example sentences.

Another example:
I will have finished editing this post by an unknown time in the future.

I will have finished editing this post by some point in the future.

To make it more definite:

It is now 0510 UTC on 05 Dec 2014. I will have finished editing this post by 0515 UTC.

Before it gets to be 0515 UTC, I will have finished editing this post.

(Using will: I will finish editing this post by 0515 UTC.)

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Time as a count nount can also mean an indefinite period of time (ODO 2.3).

Here is a famous English text that illustrates time as a count noun. In this case, time means indefinite period of time (see ODO for synonyms such as interval, stretch, season.) I include it, because of it shows over and over time preceded by the indefinite article. :)

Ecclesiastes 3 King James Version (KJV)

1 To every thing there is a season, and a time to every purpose under the heaven:

2 A time to be born, and a time to die; a time to plant, and a time to pluck up that which is planted;

3 A time to kill, and a time to heal; a time to break down, and a time to build up;

4 A time to weep, and a time to laugh; a time to mourn, and a time to dance;

5 A time to cast away stones, and a time to gather stones together; a time to embrace, and a time to refrain from embracing;

6 A time to get, and a time to lose; a time to keep, and a time to cast away;

7 A time to rend, and a time to sew; a time to keep silence, and a time to speak;

8 A time to love, and a time to hate; a time of war, and a time of peace.

And here is a popular song that uses these lyrics:

The Byrds - 'Turn, Turn, Turn' (on youtube)

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'

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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.

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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