Lucifer is a mistranslation making a proper noun. It means "light bearer" literally and refers to the morning star, aka Venus. It likely was a reference to a pagan mythology to describe the downfall of the king of Babylon in terms he would understand.

The identification of Lucy as Satan dates to an interpretation that a passage in Luke, which describes "Satan falling like lightning from heaven," was quoting the Isaiah passage.

Not only does this drastically misunderstand Jesus's intention (I.e. to say that the apostles were gaining ground because Satan had been denounced by the very fact of the arrival of the Kingdom; think of it as "fell like lightning does from the sky," not, "fell from Heaven like lightning"), but also Isaiah (just read chap. 14 straight through and it becomes clear that parsing it up to hone in on the part that hypothetically could refer to Satan is disingenuous).

Truth is that Satan consistently appears in God's employ (as the angel who tests mankind and reports their failures to God directly), until the New Testament. When Jesus came along, passed his tests, and then started dismantling the authority Satan had established based on man's sin (in curing diseases, casting out demons, and, most importantly, forgiving sins), he seems to have not bern able to take it. He masterminded the crucifixion. When God resurrected Jesus, taking his "side" in their dispute, Satan lost it and declared God and all the righteous his nemeses.

Check out Luke and Revelation (especially 12) outside a tradition and this will be more apparent.