I know that the explanation of Saul's death from 1 Samuel 31 contradicts what the Amalekite told David in 2 Samuel 1, but how do we even have the first explanation? Both people committed suicide according to the text, so who could have be the witness? The Philistines seem to have found his body first (also included in 1 Sam 31).
There was no mention of a third person with Saul and the armor bearer, so how would the writer come to the conclusion that the events happened in that way?
The answer is in 1 Samuel chapter 31, the next verse on from Saul, then his armour-bearer, committing suicide.
Saul had been critically wounded with arrows. Saul told his armour-bearer to draw his own sword and run him through. He would not do it (no doubt terrified to kill the Lord's anointed one.) Saul then fell upon his own sword and died by his own hand. The armour-bearer then did likewise with his own sword, so both men died by their own hand using their own respective swords.
Anybody coming upon the scene thereafter would see two men impaled on their own swords. The king's sword would be distinctive, unique, even. It would be obvious that the king had fallen upon his own sword (likewise the other man).
Then verse 7 tells us:
"When the Israelites along the valley and those across the Jordan saw that the Israelite army had fled and that Saul and his sons had died, they abandoned their towns and fled." N.I.V.
There were Israelite witnesses to the route and the final outcome. That is why David knew the Amalekite was lying. Others who saw the scene of suicide had testified, giving the details, which did not square with the Amalekite's story. It was not necessary to have witnessed the two suicides to work out who had done what. Basic forensic stuff.
The reports of how Saul died in 1 Samuel 31 and 2 Samuel 1 are un-related, that the former is true and the later is fabricated.
The death of Saul in 1 Samuel 31 did have witnesses.
7 When the Israelites along the valley and those across the Jordan saw that the Israelite army had fled and that Saul and his sons had died, they abandoned their towns and fled. And the Philistines came and occupied them.
11 When the people of Jabesh Gilead heard what the Philistines had done to Saul,
12 all their valiant men marched through the night to Beth Shan. They took down the bodies of Saul and his sons from the wall of Beth Shan and went to Jabesh, where they burned them.
Saul as the king of Israel, should have his royal soldiers protected him on escape. Though vv6 told 'and all his men' but this phrase didn't exist in the Septuagint. In vv11, if the people of Jabesh Gilead heard Saul was dead, it had to be from Saul's refugees rather than from the Philistines.
How did David know the Amalekite fabricated the story?
4 Saul said to his armor-bearer, “Draw your sword and run me through, or these uncircumcised fellows will come and run me through and abuse me.” (1 Samuel 31:4 NIV)
David should have known Saul, if Saul would not kill by uncircumcised fellow, how would he asked an Amalekite to kill him?
8 “He (Saul) asked me, ‘Who are you?’, “‘An Amalekite,’ I answered.
9 “Then he said to me, ‘Stand here by me and kill me! I’m in the throes of death, but I’m still alive.’ (2 Samuel 1:8-9 NIV)
David killed the Amalekite not because he did kill Saul, but because he claimed himself killed Saul
16 For David had said to him, “Your blood be on your own head. Your own mouth testified against you when you said, ‘I killed the Lord’s anointed.’” (2 Samuel 1:16 NIV)
The narrator wished to show David as God's representative who honored the position of King Saul (the "Lord's anointed,") so he portrays the Amalekite as lying about incident to gain a reward from David.
David said to the youth who was reporting to him, “How do you know that Saul and his son Jonathan are dead?” 6 The youth reporting to him replied: “I happened to find myself on Mount Gilboa and saw Saul leaning on his spear, with chariots and horsemen closing in on him... 9 Then he said to me, ‘Stand over me, please, and put me to death, for I am in great suffering, but still alive.’ 10 So I stood over him and put him to death, for I knew that he could not survive his wound. I removed the crown from his head and the armlet from his arm and brought them here to my lord.”
The report does not quite match the one in 2 Sam. 31, where Saul's armor bearer commits suicide along with Saul after seeing that the king had died. But the two reports do agree that Saul attempted to end his own life. The question arises as to whether the armor-bear was mistaken in thinking Saul was dead, or the Amalekite youth was lying in hope that David would reward him for his supposed role in his rival's death. Being a supporter of David's kingship, narrator implies that the Amalekite was lying but accepts the idea that Saul had intentionally fallen on his own weapon. He takes David's interpretation as being correct when he declares to the Amalekite:
“Your blood is on your head, for you testified against yourself when you said, ‘I put the Lord’s anointed to death.’”
So the author of 1 Samuel knew of Saul's (attempted) suicide from the Amalekite's report and David's response. The narrator, a supporter of David, casts the Amalekite as embellishing the story to seek reward from David.
