Who killed Saul, Saul or the Amalekite?

Who killed Saul, Saul or the Amalekite? 
1 Samuel 31:4 and 2 Samuel 1:8-10

 

1. Saul did (1 Samuel 31:4) - "Then Saul said to his armor bearer, "Draw your sword and pierce me through with it, lest these uncircumcised come and pierce me through and make sport of me." But his armor bearer would not, for he was greatly afraid. So Saul took his sword and fell on it."
2. The Amalekite did (2 Samuel 1:8-10) - "And he said to me, ‘Who are you?’ And I answered him, ‘I am an Amalekite.’ ⁹"Then he said to me, ‘Please stand beside me and kill me; for agony has seized me because my life still lingers in me.’ ¹⁰"So I stood beside him and killed him, because I knew that he could not live after he had fallen. And I took the crown which was on his head and the bracelet which was on his arm, and I have brought them here to my lord."

     1 Samuel 31:4 gives what actually happened while 2 Samuel 1:8-10 only gives what the Amalekite said happened.  Most probably, the Amalekite took the opportunity to benefit from the King's death, gathered his crown and bracelet and then brought them to David.  Unfortunately for the Amalekite, David said in 2 Samuel 1:13-16, "And David said to the young man who told him, "Where are you from?" And he answered, "I am the son of an alien, an Amalekite." ¹⁴Then David said to him, "How is it you were not afraid to stretch out your hand to destroy the Lord’s anointed?" ¹⁵And David called one of the young men and said, "Go, cut him down." So he struck him and he died. ¹⁶And David said to him, "Your blood is on your head, for your mouth has testified against you, saying, ‘I have killed the Lord’s anointed.'"
     The Amalekite probably thought he'd benefit from bringing the King's possessions to David, but his plan backfired.