Saul rose up and went out of the cave to continue his search, when a voice fell upon his startled ears, saying, “My lord the king.” He turned to see who was addressing him, and lo! it was the son of Jesse, the man whom he had so long desired to have in his power that he might kill him. David bowed himself to the king, acknowledging him as his master. Then he addressed Saul in these words: “Wherefore hearest thou men's words, saying, Behold, David seeketh thy hurt? Behold, this day thine eyes have seen how that the Lord hath delivered thee today into mine hand in the cave: and some bade me kill thee; but mine eye spared thee; and I said, I will not put forth mine hand against my lord; for he is the Lord's anointed. Moreover, my father, see, yea, see the skirt of thy robe in my hand: for in that I cut off the skirt of thy robe, and killed thee not, know thou and see that there is neither evil nor transgression in mine hand, and I have not sinned against thee; yet thou huntest my soul to …
"David also arose afterward, and went out of the cave, and cried after Saul, saying, My lord the king. And when Saul looked behind him, David stooped with his face to the earth, and bowed himself."
The EGW passage describes David's address to Saul after leaving the cave, including his bowing and calling Saul "My lord the king." The verse from 1 Samuel 24:8 directly supports this description, making it an explicit allusion.
"When Saul heard the words of David he was humbled, and could not but admit their truthfulness. His feelings were deeply moved as he realized how completely he had been in the power of the man whose life he sought. David stood before him in conscious innocence. With a softened spirit, Saul exclaimed, “Is this thy voice, my son David? And Saul lifted up his voice, and wept.” Then he declared to David: “Thou art more righteous than I: for thou hast rewarded me good, whereas I have rewarded thee evil.... For if a man find his enemy, will he let him go well away? wherefore the Lord reward thee good for that thou hast done unto me this day. And now, behold, I know well that thou shalt surely be king, and that the kingdom of Israel shall be established in thine hand.” And David made a covenant with Saul that when this should take place he would favorably regard the house of Saul, and not cut off his name. PP 662.2"
"And he said to David, Thou art more righteous than I: for thou hast rewarded me good, whereas I have rewarded thee evil."
The EGW passage directly quotes and elaborates on the biblical text, making it an explicit allusion rather than a thematic parallel.
"After evil-minded men have engaged in doing and saying wicked things against the Lord's servants, the conviction that they have been in the wrong sometimes takes deep hold upon their minds. The Spirit of the Lord strives with them, and they humble their hearts before God, and before those whose influence they have sought to destroy, and they may change their course toward them. But as they again open the door to the suggestions of the evil one, the old doubts are revived, the old enmity is awakened, and they return to engage in the same work which they repented of, and for a time abandoned. Again they speak evil, accusing and condemning in the bitterest manner the very ones to whom they made most humble confession. Satan can use such souls with far greater power after such a course has been pursued than he could before, because they have sinned against greater light. PP 662.4"
"But it is happened unto them according to the true proverb, The dog is turned to his own vomit again; and the sow that was washed to her wallowing in the mire."
Both passages describe a return to sinful behavior after a period of repentance. The EGW passage elaborates on the process, while the verse from 2 Peter uses a proverb to illustrate the same principle.