Why do people interpret "Lucifer" as being Satan in Isaiah 14?

I think your original question (before it was edited) makes more sense. How do you ever know WHY "others" believe what they believe? Notwithstanding, I can only answer as to why I agree with you that the frequent interpretation of many people does not square with Scripture for the reasons you state, and many other reasons (although admittedly my own interpretation). Yet the reasons to reject the interpretations of the "theysayers" (as my wife would say) are sufficiently too numerous and profound to set them out in this answer.

Accordingly, as to this particular "other" (me), I can only point to the portion of a paper that I wrote specifically dealing with the issues you raised. I ran into the same problem that you did many years ago. Beginning on page 19 of that paper, under "The Beast—the resurrected Lucifer—that eighth horn of Babylon, as extended", you will see that clear reading of the Scripture ties that king of Babylon--as extended-- to the eighth horn of Revelation at that future time of the resurrection of, and the judgment of antichrist. Of course, you have my welcoming invitation to read the paper from the beginning to be better prepared for this portion concerning Lucifer.

ADDENDUM BY EDIT TO PROVIDE "IN ANSWER REFERENCES":

According to the request of one of the site monitors, or one having authority do request such, I will copy and paste the following portion of my paper that I cited above for reference:

The Beast—the resurrected Lucifer—that eighth horn of Babylon, as extended

That is exactly what the Word of God clearly teaches about Lucifer—a man—who is somehow mysteriously and falsely depicted as being Satan by many religionists. Chapter 14 of Isaiah is not talking about Satan the fallen angel, but rather quite specifically about a “man” who will be cast out of his grave “like an abominable branch”—that vile wicked resurrected king of Babylon, extended—the beast of Daniel chapter two, extended.

This is not a history of the past as all the Lucifer-equals-Satan folks claim. This period of time only occurs when the captivity by the little horn of Daniel is finally brought to a sudden end—when the children of Israel are dwelling safely in their own land with Messiah as their king. Isaiah 14:1-3 is very specific concerning both who the LORD is speaking to, and the exact time when the future fulfilment of this prophecy will take place:

For the LORD will have mercy on Jacob, and will yet choose Israel, and set them in their own land: and the strangers shall be joined with them, and they shall cleave to the house of Jacob. And the people shall take them, and bring them to their place: and the house of Israel shall possess them in the land of the LORD for servants and handmaids: and they shall take them captives, whose captives they were; and they shall rule over their oppressors.

And it shall come to pass in the day that the LORD shall give thee rest from thy sorrow, and from thy fear, and from the hard bondage wherein thou wast made to serve, That thou shalt take up this proverb against the king of Babylon, and say, How hath the oppressor ceased! the golden city ceased! (My emphasis)

This is not describing that historical event concerning the battle of the angels spoken of in Revelation 12:7-11 which tells about Satan and his angels actually having been cast out of heaven by Michael and his angels. This is clearly a prophecy of Isaiah setting the future specific time for the children of Israel to take up a certain proverb together with its chiding against that “wicked” king of Babylon, extended—that “wicked” little horn of Daniel, chapter seven.

This narrative is a resounding chiding that depicts Lucifer as a “feller”—a lumberjack —one who fells, or cuts down the trees of the nations—including all the fir trees and the cedars of Lebanon. Remember, trees are of the earth. Trees typify humanity. At that future time this “feller” will have already himself been cut down to the ground like a fallen tree. He himself, being only a man, has also been felled just like all those kings that he cut down, as we see in Isaiah 14:12:

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! (My emphasis)

Lucifer is also typified as the “son of the morning”—Venus—which is at its brightest just before the day star—the sun—arises. The day star is Jesus Christ who returns to earth to rule just at the time that Lucifer is at his height of power. The “man,” Lucifer never was in heaven. He will only say in his heart that he “will ascend” into heaven, as we find continuing in verses 13-14:

For thou hast said in thine heart, I will ascend into heaven, I will exalt my throne above the stars of God: I will sit also upon the mount of the congregation, in the sides of the north: I will ascend above the heights of the clouds; I will be like the most High.

In verses 15-18, this “man” will have himself been brought down to hell, to the sides of the pit.

Yet thou shalt be brought down to hell, to the sides of the pit. They that see thee shall narrowly look upon thee, and consider thee, saying, Is this the man that made the earth to tremble, that did shake kingdoms; That made the world as a wilderness, and destroyed the cities thereof; that opened not the house of his prisoners? (My emphasis)

Verses 18-19 show that Lucifer will be a resurrected man who is cast out of his grave.

All the kings of the nations, even all of them, lie in glory, every one in his own house. But thou art cast out of thy grave like an abominable branch, and as the raiment of those that are slain, thrust through with a sword, that go down to the stones of the pit; as a carcase trodden under feet.

This is not saying that Lucifer initially was “slain, thrust through with a sword.” Rather, he joins the remnant of those so slain, thrust through with a sword. Isaiah, verses 20-21 reveals the exact condition that we saw of the beast, along with the false prophet who will not be joined with those kings of the nations in burial when Christ defeats them.

Thou shalt not be joined with them in burial, because thou hast destroyed thy land, and slain thy people: the seed of evildoers shall never be renowned. Prepare slaughter for his children for the iniquity of their fathers; that they do not rise, nor possess the land, nor fill the face of the world with cities. (My emphasis)

Satan is a spirit, not a man. Spirits have no carcass to be resurrected or to be trodden under feet. Satan is clearly never shown to be resurrected. But Lucifer and his false prophet, will be “cast alive” as resurrected “men” into the lake of fire. Notice that this “man” is called an “abominable branch,” but Jesus is that “man whose name is the BRANCH” (Zech. 6:12) , that son of David (Isa 11:1) , who is the “righteous Branch” (Jer. 23:5-6) .

Those men who are named in the lineage of the “seed of the woman” especially that singular seed—Jesus Christ—became “mighty men of old, men of renown” . But this antichrist is called “the seed of evildoers” who “shall never be renowned.” Titus never had any sons to inherit his emperorship after he died and he will not have a son in his resurrection. Matthew 22:30 states:

For in the resurrection they neither marry, nor are given in marriage, but are as the angels of God in heaven.

No one has, or will inherit the kingdom of that “little horn.” Even though others of decreasing importance—such as his brother, Domitian—ruled briefly as lessor Roman emperors after Titus died, no one else has ever stood in the estate of Titus. However, Titus will return to rule in his own world-order estate during the tribulation. He will also stand in the estate of that wounded world-order governmental body which is that head of the beast that was “wounded to death”. That head that was wounded to death will at that time be resurrected—because its “deadly wound will be “healed” (Rev. 13:1-3).

I know that Ezekiel and Isaiah are two different authors, but were they not of the same period of time? Ezekiel writes a similar type of passage about the King of Tyre, where he describes Tyre as the Guardian Cherub (from the Garden of Eden)

Ezekiel 28:12–15:

“‘You were the seal of perfection, full of wisdom and perfect in beauty.
13 You were in Eden, the garden of God; every precious stone adorned you: carnelian, chrysolite and emerald, topaz, onyx and jasper, lapis lazuli, turquoise and beryl.
Your settings and mountings were made of gold; on the day you were created they were prepared.
14 You were anointed as a guardian cherub, for so I ordained you.
You were on the holy mount of God; you walked among the fiery stones.
15 You were blameless in your ways from the day you were created till wickedness was found in you.

Does this not sound similar to the way Isaiah speaks of the King of Babylon? Yet clearly Ezekiel, who was writing in a similar time period, is describing this King telling him that he was in Eden and that he was anointed the guardian cherub. I assume, but I may be wrong, that this Cherub is the one with the flaming sword. By the way, the Hebrew word for flaming is lahat, which means to hide by use of the occult.

So does this not describe a fallen angel? So is it not possible that Isaiah also had a fallen angel in mind and that perhaps Isaiah and Ezekiel had a pretty clear conception of our modern day Satan.

Isaiah did not have the understanding of Satan that is commonly held today. You are correct that in the original context the passage is actually about the king of Babylon. Demonology really did not rise up until the Maccabean era, so to say that Isaiah was first writing with Satan in mind would be inaccurate (seeing as the language is more referencing the Tower of Babel). Isaiah 14 is a passage that has been looked back upon and interpreted in light of people's views of Satan to the loss of remembering the original context.

For more on the subject check out the Isaiah commentary put out by Westminster press! I hope this helps!

It is commonly believed to be Satan. "Lucifer", however, is a transliteration of the word in the Latin Vulgate. So, basically, it's a transliteration of a translation. Many more modern translations render the original word rather than following the traditional transliteration.

How you are fallen from heaven, O Day Star, son of Dawn! How you are cut down to the ground, you who laid the nations low! (Isaiah 14:12, ESV)

quomodo cecidisti de caelo lucifer qui mane oriebaris corruisti in terram qui vulnerabas gentes (Isaiah 14:12, Biblia Sacra Vulgata).