Calling Satan by the name “Lucifer” is not just an LDS issue. Many Christians have had it in their head for centuries that Lucifer is the name of Satan. This notion stems from the only place the word Lucifer is used in the Bible. In the King James Version of the scriptures, Isaiah 14:12 says, "How art thou fallen from heaven, O Lucifer, son of the morning! How art thou cut down to the ground, which didst weaken the nations!"
First, the word 'Lucifer' is Latin, so why would it be in Isaiah, which was written in Hebrew? The actual Hebrew word used in the manuscripts is 'heylel'. This word has a meaning similar to 'shining one' or 'son of the morning.' The word 'lucifer' came into play when Jerome translated the Old Testament into the Latin Vulgate in approx 400 A.D.. In Latin, lucifer means light-bearing, morning star, or the planet Venus. Now, the KJV was not translated into English from Hebrew texts, but largely from other existing translations such as the Latin Vulgate. When the translators arrived at this section of Isaiah, for some reason they never actually translated the word from Latin into English.
Now that we know where the word came from, we must tackle the issue of context. Those who attribute Lucifer to Satan interpret Isaiah 14 to be speaking about Satan falling from Heaven. However, the context does not support this idea. Isaiah 14:3 says “When the Lord has given you rest from your pain and turmoil and the hard service with which you were made to serve, you will take up this taunt against the king of Babylon.” The context is regarding the King of Babylon, oppressor of Israel. God had promised deliverance for Israel. The taunt is that the Babylonian king, though once mighty and glorious ruling over earth (like a bright star), would be cut down to the ground in humility. You can read about the fulfillment of this prophecy in the book of Daniel. It would be very odd for a teaching about Satan to be inserted at random in the context of Isaiah 14; that interpretation is simply not a logical option.
The false notion that Lucifer is the name of Satan remains a common belief. This does not present a huge doctrinal issue in mainstream Christianity. A misunderstanding amongst some of a faith does not nullify the faith in its entirety. However, when an idea that has been proven false remains a permanent and central doctrine amongst even leaders of a faith, that does present an issue. If we are going to say that the scriptures of God are true, and then use ideas that have been proven wrong to support ourselves, then we only look foolish and prove our faith is false.
Joseph Smith, the first prophet of the LDS church, supported the myth of Lucifer and continued to add on to the story. In the Book of Mormon, chapter after chapter of Isaiah has been copied over. 2 Nephi 24 is where Isaiah 14 has been copied. The Book of Mormon is supposed to be a translation of Hebrew (and ‘Reformed Egyptian’, whatever that is), but in 2 Nephi 24:12 the Latin word Lucifer is used as a proper name. Also, each chapter of the Book of Mormon has a heading containing a synopsis of the chapter. In the synopsis for 2 Nephi 24, it reads “...Lucifer was cast out of heaven for rebellion...” That is NOT what Isaiah said!
After the Book of Mormon, Joseph Smith wrote Doctrine and Covenants, a book the LDS church holds as scripture. In D&C 76:25-27 it reads, “And this we saw also, and bear record, that an angel of God who was in authority in the presence of God, who rebelled against the Only Begotten Son whom the Father loved and who was in the bosom of the Father, was thrust down from the presence of God and the Son, And was called Perdition, for the heavens wept over him—he was Lucifer, a son of the morning. And we beheld, and lo, he is fallen! is fallen, even a son of the morning!”
Unlike many early teachings of the LDS church that have since been abandoned, this teaching is still sustained. In a November 1987 Ensign article by LDS apostle James E. Faust titled “The Great Imitator”, Faust refers to Satan multiple times as Lucifer. And, also found on LDS.org, the Bible Dictionary says the following of Lucifer: “Apparently Lucifer is the name of the devil before his rebellion and fall. Latter-day revelation clarifies the fall of Lucifer and equates him with Satan.”
Apparently, no one has read the Bible without their LDS glasses on. A mistake is one thing, but why would the LDS church still perpetuate a known mistake as established doctrine when they are supposed to be the mouthpieces of God? Like the apostle Paul said, “what I am doing I will continue to do, in order to undermine the claim of those who would like to claim that in their boasted mission they work on the same terms as we do. For such men are false apostles, deceitful workmen, disguising themselves as apostles of Christ. And no wonder, for even Satan disguises himself as an angel of light. So it is no surprise if his servants, also, disguise themselves as servants of righteousness. Their end will correspond to their deeds.” - 2 Corinthians 11:12-15.