The King James Bible advocates, in their attempt to defend '1 Samuel 13:1' said:
It is generally believed that the historical books of the Bible (Samuel, Kings and Chronicles) either made use of earlier sources or are composite texts of those originally separate sources (the book of Jasher (2 Samuel 1:18), the chronicles of the kings of Israel (1 Kings 14:19), the chronicles of King David (1 Chronicles 27:24), the book of Samuel the seer, the book of Nathan the prophet, and the book of Gad the seer (1 Chronicles 29:29)). Therefore different scribes may have written different parts of the book of 1 Samuel. Perhaps Samuel was the scribe who penned 1 Samuel 13:1. The events described fall during his lifetime. This portion written by Samuel may in fact be from the book of Samuel the seer referred to in 1 Chronicles 29:29. The scribe of 2 Samuel 2:10 (the first use of the regnal formula) and onward had to be someone other than Samuel because Samuel had died by the time of 1 Samuel 25:1. Samuel may have used the phrase with "בן שנה" and "במלכו" differently from the later scribes who used the phrase as part of the regnal formula.
What to take away from the above is that
1. The Bible is not inspired by God. Instead, it is written from a composite of earlier writings.
2. The real authors of the Biblical books are unknown. For example, the scribe of 2 Samuel 2:10 and onward had to be someone other than Samuel because Samuel had died by the time of 1 Samuel 25:1. Yet, no one can tell you who the scribe of this 2 Samuel 2:10 is for sure.
The page continued to say the following:
Prior to the establishment of a monarchy in Israel, there would not have been such a thing as a regnal formula. Having no king, there would have been no need for a formula for introducing a king. Samuel most likely learned how to write during his youth. Yet during his youth there was no such thing as a monarchy in Israel and probably no such thing as a regnal formula. Even if the use of the regnal formula became current during Samuel's lifetime, Samuel could have still used language in the way he had learned in his youth. Even today, we can see the older generation using language in such a way that the mainstream generation might deem archaic. Perhaps Samuel at 1 Samuel 13:1 used the phrase in question in its ordinary literal meaning as he had learned in his youth. On the other hand, the later generation of scribes from 2 Samuel 2:10 onward were familiar with the regnal formula so they used it. When the source materials were put together to form the book of 1 Samuel as we have today, the editor did not edit the outdated use of the phrase at 1 Samuel 13:1.
The implication of the above is that the author of 1 Samuel 13:1, whoever he is, is not writing under the inspiration of God. This is because if he were, then God would have inspired the reignal formula to him even if he did not know it.