A contradiction on Saul's death, and an unknown book

The death of Saul, by Gustave Dore.  GNU License

2 Samuel, Chapter 1 - Second Samuel begins with David coming out of hiding after learning about the death of King Saul, who had been hounding him since he learned that Yahweh (aka God) favored David more than he.  

Whereas in First Samuel Saul died by falling on his sword (pictured above), in this chapter David happened upon an Amalekite who had a different story.  The Amalekite confessed that he killed Saul after the king fell on his spear.  King Saul, in pain, saw the Amalekite and basically asked him to finish him off.  I know, that's not a big issue, it sounds more like a plot twist if anything.  But there are other accounts of Saul's death, where he is killed by a Philistine or even by God Himself.

Another interesting subject in this first chapter is an introduction to a book that must have existed then, but no longer exists - the Book of Jasher.  Translated, Jasher means "just man", so it is actually titled "Book of the Just Man".  That actually sounds like it would be an interesting read!  

There are books called "Book of Jasher" which have copies that date much later than the dates Second Samuel is thought to have been written (between 640 to 530 BCE), but here is one possible digital copy of the book.  The scrolls which this one may have been translated from dates to about 1613, just a few centuries ago.

I am still issues with the photos, but I think I may have found a workaround.  

Who wrote 1st Samuel, Chapter 31?

Saul falls upon his own sword in this book (1st Samuel).  Stay tuned to see how he dies in 2nd Samuel!

(1 Samuel, Chapter 31) In this final chapter of 1st Samuel, the numerically superior Philistines do battle with their old enemies the Israelites, of whom King Saul is their commander.

According to the book, the battle was very one-sided in the favor of the Philistines.  King Saul took numerous arrows and asked his armor bearer to go ahead and kill him, but he would not.  So, Saul took a sword and fell upon it, thereby killing him.  Pay attention to this detail, because the first chapter of the next book (2nd Samuel), we find a different story of Saul's death.

Anyhow, I still think that the authors of the book chose to end it like this.  I'm not sure how historical this story is and how mythical it is.  None the less, I would have to say that it was written very weird.  The redundancies and the seeming inability of characters to act on what happened in previous chapters, give the impression that 1st Samuel had multiple authors, or multiple manuscripts, which were later just cobbled together.

Who wrote 1st Samuel?  Tradition says that Samuel was the one wrote it, but like Moses and the Pentateuch, Samuel's character dies in this book. He dies specifically in Chapter 25.  Another theory is that multiple 'prophets', including Samuel, had a hand in its authorship.  The other two prophets listed as contributors to 1st Samuel are Gad and Nathan.  

More than likely, this book did have numerous authors and also utilized court records and documents from the Israelite nation.