Talut (Arabic: طالوت, Ṭālūt) is considered to be the Qur’anic name for Saul, as he was the Malik (Arabic: مَـلِـك, King) of Israel, or Gideon, with the reasoning that the Quran references the same incident of the drinking from the river as that found in the Book of Judges (7:5-7), and other factors associated with the latter.
Thank you for the interesting question!
As you know, there are a number of differences between the Qur'anic and Biblical stories of similar people. Muslims and those who believe in a literal interpretation of the Old Testament will have to agree to disagree until the day comes when the truth is made clear!
In this case, it is more complicated because it is not 100% clear that Talut is the same person as Saul because they have different names.
There is a good discussion of the story in the relevant section of The Study Quran, ed. S. H. Nasr. Also here are some hadith on the subject. https://www.al-islam.org/hayat-al-qulub-vol-1-allamah-muhammad-baqir-al-majlisi/account-ishmaeil-talut-and-jalut
By the nature of oral literature, when a story is told, it is often altered or embellished, and this is not considered a problem in cultures that transmit their literature and history orally. The oral retelling of historical events still continues in many parts of Africa, almost as a spiritual ritual. The genealogist, or story-teller, will claim to have heard the story from his father, who heard it from his father, and so on. There is meant to be a chain of transmission which should protect against error; nevertheless, while you can find the same basic framework of the narrative in different accounts, there will often be notable differences.
The narratives of the Bible were compiled over time by different groups of scribes. Initially, narratives from the Bible were transmitted orally - even for several generations. Eventually they came to be written down. The accounts that have been written in the Bible may not be the only accounts that exist of certain narratives. There may be several variations.
In fact, often certain points that the Qur'an makes about particular doctrines held among Jews or Christians actually refer to apocryphal beliefs that were not held by the majority. There were often sects and sub-sects in Arabia and Iraq and not much has been written about them.
The Holy Quran describes the story in Chapter 2, Surah Baqarah, from verse 246 to 251. We believe the Qur'an is the word of God which is unchanged and unfabricated and 100% true. However the bible has gone through much distortion and fabrication. Therefore the story of Talut in the Quran is the accurate version.
There are historic references in hadith litrature but scholars are careful in taking these as they have to match the Quran.
Here is a lecture on Talut that I gave that you may benefit from: