Disillusioned there are a few areas. David is of course Saul's son-in-law in the current narratives. However David has three introductions.
1 Samuel 16 that you quoted I feel is late and Deuteronomist having Samuel anoint David among his brothers as one of Jesse' sons.
There are two more 1 Samuel 17.has two narratives and both are amusing:
1 Samuel 17:12-15 seems to state the obvious but watch my highlighting>>...Now David was the son of that Ephrathite of Bethlehem Judah, whose name was Jesse; and he had eight sons. The man was an elderly old man in the days of Saul. 13 The three oldest sons of Jesse had gone after Saul to the battle; and the names of his three sons who went to the battle were Eliab the firstborn, and next to him Abinadab, and the third Shammah. 14 David was the youngest; and the three oldest followed Saul. 15 Now David went back and forth from Saul to feed his father’s sheep at Bethlehem.- WEB
The texts twice repeats the three sons and introduces Jesse into the narrative as a side note. ;The original likely only had the bold. (My theory)
I need to mention that verse 12-31 are not original to the LXX translation. Even the NWT 1984 reference bible has a footnote.
Not after verse 31 David and Saul are in an argument and the if we removed David's Greek tale of slaying lions the text seems to indicate David and Saul have a close relationship. This whole next several verses contradict chapter 16 where the Deuteronomist says David is a man of war. and an armor bearer Here he is a child who has never used armor.
Note however the third introduction to David by the idiot trying to merge the accounts in verse 54-58 in the World English Bible with my italics
David took the head of the Philistine and brought it to Jerusalem, but he put his armor in his tent.
(What a fool to add this verse^^^ David would not conquer Jerusalem for awhile)
55 When Saul saw David go out against the Philistine, he said to Abner, the captain of the army, “Abner, whose son is this youth?”
(If we assume the Deuteronomist is thrusting David into a pervious story then this makes sense. But in this narrative it become laughable and Saul in need of medical help.)
Abner said, “As your soul lives, O king, I can’t tell.”
56 The king said, “Inquire whose son the young man is!”
57 As David returned from the slaughter of the Philistine, Abner took him and brought him before Saul with the head of the Philistine in his hand.
(That was a worthless journey to Jerusalem with a rotting head uphill and a downhill hike back to Saul. This is so forced onto the account it ludicrous.)
58 Saul said to him, “Whose son are you, you young man?”
David answered, “I am the son of your servant Jesse the Bethlehemite.”
(Duh!)
As for a direct reference the problem is that 1 Samuel 18 continues this narrative however there is a massive doublet later with David sparing Saul's life twice. The oldest one says:
It came to pass, when David had finished speaking these words to Saul, that Saul said, “Is that your voice, my son David?” Saul lifted up his voice, and wept. -1 Samuel 24:16 WEB
The other version uses the previous thus at 1 Samuel 26:17-
17 Saul recognized David’s voice, and said, “Is this your voice, my son David?”
There is obviously a text that has been reworked and in some versions like the latter copies of the Septuagint even removed a large chunk of the David and Goliath story to harmonize things.
So the original question in the original J story (Yahwist) was David Saul's son. The answer is simple. I have no clue! LOL. Actually I am still researching this but I find the three introductions to David and Jesse very intrusive and will be following that lead eventually.