In the Old Testament, "Satan" generally has a different meaning from what we are used to. It is actually a transliteration from a Hebrew word meaning enemy, adversary or accuser. Although Old Testament names usually have a specific meaning, satan is not used as a personal name, but rather a generic label for an enemy. In other words, the Old Testament does not use "Satan" with capital 'S' to describe a specific being that "rebelled against [God]" and was "cast down" (Moses 4:3). Frequently it is used to describe humans. In Numbers 22:22, 32, even God's angel is a "satan" functioning as a threatening enemy to a pagan magician trying to harm Israel. Only occasionally it is used to describe heavenly beings and few details are given.
Many Bible scholars would claim that our modern understanding of Satan developed after Old Testament times. However, the Book of Moses in the Pearl of Great Price makes it clear that Adam, Enoch and Moses had an understanding similar to ours so this might also be a case of "plain and precious things" being lost. In that regard, Moses 4:1 is particularly interesting.
And I, the , spake unto Moses, saying: That , whom thou hast commanded in the name of mine Only Begotten, is the same which was from the , and he came before me, saying—Behold, here am I, send me, I will be thy son, and I will all mankind, that one soul shall not be lost, and surely will do it; wherefore me thine honor.
Even though Satan in this verse is naturally spelled with capital 'S' in English, the wording suggests an underlying Hebrew understanding. The verse is referring to Moses 1, where Moses is confronted by Satan and his first reaction is asking him "who art thou?" (Moses 1:13). In Hebrew, Moses would have been confronted by "enemy". Then, after commanding him to leave in the name of the Only Begotten, he talked with God again who gave him the creation account and then explained who "that satan" (enemy) really was. Moses 4 goes on to explain that it was the same being that was in the grand council in the beginning, who sought the Father's glory, who was cast down, became the father of all lies and beguiled Adam and Eve in the garden. It would seem at that moment, at least for Moses, that "that satan" became Satan with capital 'S'. This whole explanation in the beginning of Moses 4 is found in the Pearl of Great price but missing in the Old Testament.
As I have written before, I would not be surprised if the Nephites had an extended version of Genesis similar to the Book of Moses on the Brass Plates. For instance, in 2 Nephi 2:16-17, Lehi refers to something that he has read that looks very similar to Moses 4:1-6
17 And I, Lehi, according to the things which I have read, must needs suppose that an of God, according to that which is written, had from heaven; wherefore, he became a , having sought that which was evil before God.
18 And because he had fallen from heaven, and had become miserable forever, he also the misery of all mankind. Wherefore, he said unto Eve, yea, even that old serpent, who is the devil, who is the father of all , wherefore he said: Partake of the forbidden fruit, and ye shall not die, but ye shall be as God, good and evil.
It is clear from the Book of Mormon that the Nephites generally had this understanding. But the ancient Hebrew meaning of "satan" also shines through. For instance, one Hebrew meaning of the word is the accuser in a court setting. We see this setting in the first couple of chapters in the Book of Job and Zechariah 3:1-2, for example. The word, "adversary" is sometimes used in the Book of Mormon. According to the Webster 1828 dictionary, this can be "an opponent or antagonist, as in a suit at law".
4 And thou seest that we know that thy plan was a very plan, as to the subtlety of the devil, for to lie and to deceive this people that thou mightest set them against us, to us and to cast us out—
5 Now this was a plan of thine , and he hath exercised his power in thee. (Alma 12)
Ironically, Alma is saying this to Zeezrom, who was a lawyer. The devil exercised his power in Zeezrom and used him as a tool in this court-like setting in Ammonihah to achieve his purposes and accuse Alma and Amulek. But in reality, as Alma points out, he was also Zeezrom's adversary.
This role of the devil and meaning of satan is contrasted by Christ who is our advocate or intercessor (see e.g. 2 Nephi 2:9, Mosiah 15:8). These opposites are also illustrated when the Book of Mormon describes Christ as the fountain of all righteousness
Behold, I will show unto the Gentiles their weakness, and I will show unto them that , hope and charity bringeth unto me—the fountain of all . (Ether 12:28)
...and the devil as the enemy (Hebr. satan) of all righteousness.
And now, my beloved son, notwithstanding their hardness, let us labor ; for if we should cease to , we should be brought under condemnation; for we have a labor to perform whilst in this tabernacle of clay, that we may conquer the enemy of all righteousness, and rest our souls in the kingdom of God. (Moroni 9:6)