Isaiah 14:12 in the original Hebrew reads:
איך נפלת משמים הילל בן שחר נגדעת לארץ חולש על גוים׃
Which, conservatively translated, means:
Oh how you have fallen from heaven, O bringer of the dawn, O son of the morning! how you were cut down to earth, who overthrew nations!
One has to be careful when reading the writings of early Christians, as well as New Testament writers, because they'll often apply Scriptures without qualification, as referring to something, whereas they are using the Scripture typologically, and not as if the fulfillment of which they are speaking is the original referent of the passage - and expect the reader to know such by the sheer fact of being familiar with the faith.
Because they believed Scripture was not only a recorded narrative of sacred history - of events - but rather that:
Hebrews 4:12 ...the word of God is living and effectual, and more piercing than any two edged sword; and reaching unto the division of the soul and the spirit, of the joints also and the marrow, and is a discerner of the thoughts and intents of the heart.
So, for example:
Matthew 2:13-15 And after they were departed, behold an angel of the Lord appeared in sleep to Joseph, saying: Arise, and take the child and his mother, and fly into Egypt: and be there until I shall tell thee. For it will come to pass that Herod will seek the child to destroy him. 14 Who arose, and took the child and his mother by night, and retired into Egypt: and he was there until the death of Herod: 15 That it might be fulfilled which the Lord spoke by the prophet, saying: Out of Egypt have I called my son.
This Scripture originally referred to Israel, the corporate body of believers in Yahweh; and Matthew clearly would have known this, and all his readers, because the verse says:
Hosea 11:1 Israel was a child, and I loved him: and I called my son out of Egypt.
God called Israel his "son" in the Old Testament, yet Matthew sees this as a prefigurement of what happened with Christ, the fulfillment of what was written so the fulfillment would be recognized as approved by God: when His most, and more, true Son would be delivered from Egypt, from the wrath of Herod - a king of new Pharoah.
And this applies to most of what are called 'types' in Scripture (i.e. where something real and historical prefigures some later reality) - they often apply better to the fullfillment than to even the historical event or person of which they were written!
1 Corinthians 9:8-11 Speak I these things according to man? Or doth not the law also say these things? 9 For it is written in the law of Moses: Thou shalt not muzzle the mouth of the ox that treadeth out the corn. Doth God take care for oxen? 10 Or doth he say this indeed for our sakes? For these things are written for our sakes: that he that plougheth, should plough in hope; and he that thrasheth, in hope to receive fruit. 11 If we have sown unto you spiritual things, is it a great matter if we reap your carnal things?
Here, it referred to not disallowing that which is earned or rightfully owed, using the example of oxen which have labored and are now rightfully, justly, owed it to eat afterwards. St. Paul says, "Does God concern himself with beasts? with oxen?" By which he obviously doesn't mean, and can't mean - be cruel to animals, but rather, "Did He invest his Holy Ghost in this Scripture only to prevent cruelty to animals, and not, if not much more so, to teach us about what is rightfully owed to servants of God - men, who are "of much more value" than they?" (Matthew 6:260).
Or, the sacrificing of Isaac, which is a kicking-oneself-in-the-head-for-not-realizing prefigurement of Calvary:
Genesis 22 Now it happened some time after these things that God tested Abraham, saying, "Abraham, listen to me! Take your son, your only son Isaac, whom you love, and go to the land of Moriah, and offer him there for a burnt offering, upon one of the mountains which I tell you to.
And rising early in the morning Abraham saddled his donkey and took his two young lads, and his son Isaac, and he split wood for a burnt offering: and he ascended to the place God had intimated to him.
And by the third day, Abraham could see the place afar off.
And Abraham said to the lads, Stay here with the donkey, while I and the lad go a little further, and there worship: and we shall return again to you.
And taking the wood for the burnt offering, Abraham placed it upon Isaac his son, and he took with him means of fire, and a knife: and both of them continued together.
And Isaac said to Abraham his father, Father, and he said, Yes my son? And he said, Look, here is something to make a fire, and the wood, but where is the lamb to be offered?
But Abraham said, God shall provide his own lamb for a burnt offering, my son. And they both continued together.
And when they came to the place God had intimated to him, Abraham built an altar, and arranged the wood, and bound his son upon the altar, over the wood.
And stretching forth hishand, Abraham took the knife with which he meant to slay his son, when the angel of the Lord called to him from heaven, saying, "Abraham!" and he replied, "Yes?"
And he said, "Do not put your hand near the lad, nor do anything to him: for now I know that you fear God, seeing as you have not spared your son—your only son—from me.
And looking up, Abraham saw behind him a lamb, caught in the thicket by the horns, and Abraham took the ram and offered him for a burnt offering instead of his son Isaac.
And Abraham called the name of that place, The Lord shall provide: and even to this day it is said, The Lord shall provide on the mountain.
Clearly this is recounting an event, in narrative form. And it doesn't refer to anything else as such, than what it records. However, it was viewed as a type of what was to come. It was seen as the Word of God, which is alive, and contains much prophecy for those with eyes to see.
Abraham was known as "Father Abraham" among the Jews - the father of themselves physically, as well as the Father of Judaism the religion. This typifies God the Father, who more perfectly fits this description.
Abraham's son, who bears the wood of his sacrifice, and carries it to the top of a mountain, prefigures Christ, who bore His cross to Calvary.
The wood, of course, prefigures the cross. Cf. Wis. 14:7.
The male sheep with its head caught in the thorns, which was the actual sacrifice, and not Isaac, prefigures Christ the "unblemished [male] lamb" (Exodus 12:5) who was crowned with thorns in mockery - "behold the Lamb of God;" (John 1:29) "God will provide his own lamb, my son" (Genesis 22:8).
The Jews believing that God was yet to provide a lamb, shows that they, by tradition, expected a Calvary-like Abrahamic sacrifice, such as the Cross, even if by the time of the Cross, they had forgotten or lost sight of or hope in it.
Notice the unsual emphasis on 'Your son, your only son, whom you love,' which clearly corresponds to Jesus' unique Sonship:
Matthew 3:17 And behold a voice from heaven, saying: This is my beloved Son, in whom I am well pleased.
John 3:16 For God so loved the world, as to give his only begotten Son; that whosoever believeth in him, may not perish, but may have life everlasting.
Notice also how, in light of the self-evident truth of Christianity, I said, "clearly refers to," even though it originally referred to something else.
This is how early Christians - the original apostles and first Jewish Christians - used and viewed Scripture.
This is what Jesus talked about on the road to Emaus (Luke 24:27, 32).
Thus, the Christian Church viewed this condemnation of the wicked king as a Scriptural type of Satan, the fallen angel, who, unlike the earthly king of Tyre, who metaphorically fell from heaven, or from the heights to which he had elevated himself or was elevated, was literal in heaven, and fell from the state of grace.
Lucifer is the Latin equivalent of Hallalel, meaning, the morning star. It is not a proper name; and only became such by association of this passage, in Latin ("Lucifer" - morning star) with Satan, to whom it was typologically applied.