As the revelation of Jesus Christ, we get confirmation in the Book of Revelation and in the NT that Satan is Lucifer, Satan is the Prince of this world, Satan is that old serpent in the garden.
For you to make the assertion that the Jews had no concept of him falling from heaven is incorrect and scripture proves it.
In the OT, Satan is mentioned by name in I Chronicales, Job, and Zechariah, and Psalm 109.6 among other things. There is nothning to identify the King of Sydon with Satan nor the King of Babylon.
The New testament is very clear that Christ considered Satan tre fallen angel also known as the devil, but that's not what the subject was about.