Good question - could take a book to answer! The key phrase which unlocks (or veils?) identity is in v. 12:
"O Lucifer, son of the morning!"
Hebrew: הֵילֵ֣ל בֶּן־שָׁ֑חַר | hêlēl ben-šāḥar
Greek: ὁ ἑωσφόρος ὁ πρωὶ ἀνατέλλων | ho heōsphoros ho prōi anatellōn
Grk trans.: the Day Star, which used to rise early in the morning
And, important in a moment, the Latin Vulgate:
lucifer qui mane oriebaris
trans: O Lucifer, who didst rise in the morning
1. "These verses seem to refer to the king of Babylon"
Just so. Which king of Babylon has vexed commentators for centuries. Suggestions ranging from Sargon II to Alexander the Great (!) have been made. If the key term hêlēl provides a key (it was used only rarely as a royal epithet),1 the it would point to Esarhaddon (the only Babylonian or Assyrian king for whom its use is attested).
Meanwhile, this is a "taunt against the king of Babylon" (Isaiah 14:4), even if we don't know for certain which one.
2. Why do some people believe that these verses are about Satan?
This, I take it, is the main question.
The chief reason is found in the New Testament, a statement from Jesus in Luke 10:18:
And he said to them, "I saw Satan fall like lightning from heaven."2
εἶπεν δὲ αὐτοῖς· ἐθεώρουν τὸν σατανᾶν ὡς ἀστραπὴν ἐκ τοῦ οὐρανοῦ πεσόντα.
The resonance with the first part of Isaiah 14:12 is clear, and so in many Bibles that provide cross-references, you will find this one included.
I am not aware of a tradition before Jesus that sees Satan in Isaiah 14:12, but there are several factors contributing to this:
- the "personification" of "the satan" [הַשָּׂטָ֖ן] (= "the a/Adversary") (with an article) comes very late - it's "the satan" in Job 1-2, Zech. 3:-12 -- so not a proper name, but a title or role designation -- but finally "Satan" only in 1 Chronicles 21:1;
- there is precious little evidence for the intepretative traditions of these texts before the time of Jesus;
- this is one of the things that makes the Septuagint (pre-Christian Greek translation of the Hebrew scriptures) so important,3 and you can see from the bit quoted above that any association of the "Day Star" (see text and translation, above) with "Satan" is absent from the Septuagint (Greek) Isaiah. Rather, it associates this "star" in a straightforward way with the "morning star" -- quite a natural translation for the Hebrew.
I will be very interested to see what other evidence people can bring to this discussion precisely at this point (update: see now the following section "On the Connection..."). There are some connections in pseudepigraphcial literature, but dating any of it is notoriously difficult, and I'm not sure what relevant material there would be prior to the New Testament.
So, it seems to make its first entry in the gospel of Luke, and with the authority of Jesus -- so that accounts for date of origin (perhaps!), and reason for its wide diffusion.
On the Origins of the Connection between Luke 10:18 & Isaiah 14:12
(This subsection was prompted by a comment by @kmote.) In the comments, attention was drawn to the Isaiah commentary by Gary V. Smith, Isaiah 1-39 (New American Commentary, 15; B & H Publishing, 2007), p. 314 n. 94. After nicely explaining the meaning of the Hebrew terms, he goes on to claim that
The Early Church Fathers Tertullian and and Gregory the Great connected the fall of Satan from heaven like lightning in Luke 10:18 with Isa 14:12, thus developing the unfounded view that 14:12 describes the fall of Satan.
This is very suggestive and helpful. Two things about it make me nervous: (i) the lumping together of Tertullian and Gregory as if they belong to the same historical horizon is odd, given they lived about four centuries apart (see links embedded in quote above); and (ii) we are given no references, sources, or citations for the claim. I might be unkind, but this looks to me as if Smith is relying on second hand evidence -- but it is a pointer worth following up all the same.
On checking the indices for the Ante-Nicene Fathers, however, I cannot turn up any discussions by either Tertullian or Gregory that substantiate Smith's claim. However, it was not a futile exercise, as it turned up three fascinating passages, two of which are directly relevant and even important for this question, and the third which is only intriguing.
The important "find" is that Origen appears to be the prime candidate to connect explicity Isaiah 14:12 and Luke 10:18. First, from De principiis (On First Principles), Bk. I.v.5 (brief citation: see link for whole passage):
Most evidently by these words is he shown to have fallen from heaven, who formerly was Lucifer, and who used to arise in the morning. For if, as some think, he was a nature of darkness, how is Lucifer said to have existed before? Or how could he arise in the morning, who had in himself nothing of the light? Nay, even the Saviour Himself teaches us, saying of the devil, “Behold, I see Satan fallen from heaven like lightning.”
This is the direct link (the smoking gun?) that joins together the two passages. The other, less explicit, passage also comes from De principiis, Bk. IV.i.22 (again, brief citation), and only offers support for the basis on which Origen could make the exegetical move:
And what is said in many places, and especially in Isaiah, of Nebuchadnezzar, cannot be explained of that individual. For the man Nebuchadnezzar neither fell from heaven, nor was he the morning star, nor did he arise upon the earth in the morning.
And finally, the third "intriguing" suggestion is an editorial note in an anonymous work (sometimes connected with Cyprian, contemporary with Origen), "Treatise on Re-baptism", in which the ancient author briefly discusses the "Get behind me..." (Matthew 16:23) passage. The editor comments:
Isa. xiv. 12. The sin of Lucifer had, very possibly, been this of rebelling against the Incarnation and the introduction thereby of an order of beings higher than himself. Hence our Lord recognised in Peter’s words the voice of the old adversary, and called him “Satan.”...
(One would like to know where the editor was getting this idea from!)
So Origen appears to be the origin (!) of this tradition. It would still be good to know Smith's source for the suggestion that Tertullian (also roughly contemporary with Origen) and Gregory (400 years later) contributed to this tradition. But I think I've gone as far as I can with this aspect of the question.
3. ...to what extent has the interpretation flourished throughout church history?
It's worth noting at the outset that this understanding is at home in Christian interpretation, and given that the line is drawn through a saying of Jesus' in the the gospel of Luke, that shouldn't be a surprise.
The Latin Vulgate, product of the fourth Century A.D., uses the word lucifer. It can simply be translated in its natural sense, "light bearer", combining the Latin lux "light" + fero "bring". But in wider usage, Latin lucifer also refers to the morning star, Venus, and the association of this most prominent of "morning stars" (to the ancients' way of thinking) was a natural connection with the "star" of Isaiah 14:12.4 This, at any rate, is where the name "Lucifer", as a proper name, comes from.
One would need to do a thorough search throughout the centuries of Christian intepretation to be certain of the adoption and diffusion of the idea that "Satan" is referred to in Isaiah 14:12. In Reformation period commentary, it was known and resisted. Franz Delitzsch, in his fine Isaiah commentary (1890; first German edition, 1875), quotes Luther to the effect that the tradition that Isaiah 14:12 referred to Satan was "insignis error totius papatus" = "a noteworthy error of the papacy" ... but he would say that. But Calvin, in fact, also noted and repudiated this identification in his Isaiah commentary as "arising from ignorance", and when commenting on Luke 10:18, he makes no mention of this allusion.
How widely did it eventually flourish? Very widely! In a negative sense, this can be seen also in its forceful rejection by Luther and Calvin -- they weren't rejecting an interpretation that was unknown to their readers, but one that they expected their audience to know.
What could account for this wide diffusion in a modern, popular context (such as OP has in mind)? A bit of speculation is warranted, and there is one good candidate (or culprit, depending on one's point of view): the Scofield Reference Bible (first appeared in 1909, and went through subsequent revisions). It makes the forthright claim that in Isaiah 14:12, "Lucifer, 'day-star,' can be none other that Satan". Given the massive popularity and huge influence of this work, my own guess (and it remains that!), is that this is an idea that Scofield disseminated -- or perhaps rather (since the idea was long known), validated and promoted.
One imagines Luther, Calvin, and a host of commentators through the ages, shuddering.
Summary - although the original Isaiah text is clearly aimed at some anonymous king of Babylon, Jesus uses these words to describe the defeat of darkness in the mission of the Seventy-Two in Luke 10:18. This association then was widely diffused.
NOTES
- According to M.-J. Seux, Épithètes royales akkadiennes et sumériennes (Paris, 1967), p. 80.
- That's ESV (and RSV, etc.). A better translation might be: "I was watching Satan fall from heaven...", since the Greek verb here is imperfect. The nuance would be that as the Seventy-Two were out on their mission, their kingdom activities were defeating Satan -- and Jesus was watching. But that might be too much to read into the tense of the Greek verb.
- That is, the Septuagint is one of our major witnesses to an understanding of the Hebrew scriptures in the pre-Christian era -- since all "translation" involves, in some measure, "interpretation" as well.
- It is interesting to see how this verse is handled by the Jewish commentators. Rashi associates the "morning star" with Venus, which is the expected understanding of this phrase, shared also by the Septuagint in the 2nd-3rd C. BC, and the Latin Vulgate in the 4th C. AD. Other commentators represented in the Miqra'ot Gedolot likewise treat this phrase with reference to the "star of the morning", i.e., Venus. N.b. the understanding of the figure of "satan" developed in Jewish tradition differently to Christianity.