The shortest answer is probably, yes. These could be two different source documents; the one would generally be attributed to J (Yahwistic) the other ascribed to E (Elohistic). However, that is way too defined, and few scholars would be comfortable with that single response. Indeed, today scholars know that 1 and 2 Samuel had a long oral tradition, many authors, and many editors that had a hand in shaping these books into the texts that we read in the Hebrew Bible.
They would also argue that authorship is not as important as trying to understand why the text reads as it does. In that vein, there are no shortages of interpretation. The most likely is that David is introduced (actually three times) in much the same way that Saul was introduced (three times). In both cases, the first intro was done in secret (see I Sam 9:1-10:16; I Sam 16:1-13); the second had a more public aspect (see I Sam 10:17-27; I Sam 16:14-23); and the third had military overtones (see I Sam 11:1-15; I Sam 17:1-58).
In the case of David, all three revolve around him being a shepherd. Yet, the accolades don't stop there. David is known to be skilled with a lyre; he was described as being "physically fit and attractive," a man of valor, a warrior, and he was prudent in speech (that could mean he was insightful or intelligent). Most importantly, however, the verses show that the hand of Yahweh is constantly in the background, maneuvering events to ensure that David would be the future king.
Most scholars will say that the texts are irreconcilable while others go to great lengths trying to harmonize them. In 16:1-13, Samuel is told to go to David's father, Jesse, in Bethlehem, though he does not know which son is to be Saul's replacement. Even though seven sons are paraded in front of him, Samuel is told that none of them will be the one. They all have to wait for David's arrival. He is anointed, signifying that Israel has a new king, but no words are spoken.
In the next story, Saul has an evil spirit that is quieted by healing music. He commands his attendants to find someone who plays well. One of the servants testifies about David, who is immediately commissioned by Saul. The irony, of course, is that Saul brings his own replacement to court. David does not intrude; he is invited by the king himself. And, as it turns out, David's healing music is very effective.
In the third segment, Saul is confounded by the Philistine army and Goliath, in particular. Because three of David's brothers are in the army, David brings food and supplies to the soldiers. While there, he hears Goliath's taunts, and responds with indignation. This gets back to Saul, who again sends for him. There is no acknowledgement on either side. Though Saul is king, David speaks first. But, the reader knows that David is the true king, so the narrator subtly reinforces that point. After David has slain Goliath, Saul inquires about David's identity.
Scholars who wish to harmonize these accounts point out that Saul only asks who David's father is. The theory goes that David was doing double-duty in Saul's court as well as on the battlefield. Additionally, Saul had many young armor bearers. In the heat of battle, Saul simply forgot whose son he was and asked to be reminded. Perhaps he wanted to especially honor the young man's father. This was nothing more than a reminder because, previously, Saul had sent communication to David's father, and would have known who he was (16:19).
Whether or not this is an appropriate reading of the text may have to be determined by the reader. The truth is, David was such an important figure that many stories circulated about him and some were incorporated into the texts.