Let me begin by explaining how and why Satan is called "Lucifer".
Now as for the name” Lucifer”. The Old Testament Hebrew says Heylel “הֵילֵל”, which can mean “shining one”, but also means “boastful one” since this word comes from the root word “halel” הָלַל, which means to shine, to make a show, to boast, and thus, be foolish.
And judging by the boastful rant that this “shining one” gives in Is. 14:13-14, it’s no wonder why he’s a “boastful one”. Now the Greek Septuagint translates 'Heylel' to Heosphoros ἑωσφόρος, meaning 'bringer of dawn'. Heosphoros is a variant name of Eosphorus (Greek Ἐωσφόρος, "bearer of dawn"), which in turn comes from Phosphorus (Ancient Greek: Φωσφόρος, "bearer of light"). Both these names are said to be the personification of the “Morning Star” in Greek Mythology. This is NOT to say that these names are from Greek Mythology, but merely that these names are considered the personification of the “Morning Star”. Both Heosphoros and Lucifer are proper names within the Greek and Latin language. Because Heosphoros is a name, the Latin Vulgate translated it to “Lucifer”, meaning “shining one, morning star, bringer of light”. Therefore, it’s perfectly acceptable to view Lucifer or “Shining one” as a name.
Now, “Morning Star” is also a divine title given to Jesus Christ in 2 Pet. 1:19 “φωσφόρος” (Phosphoros-morning star), and in Rev. 22:16 “ἀστὴρ ὁ λαμπρὸς καὶ ὀρθρινός” (Bright and Morning Star).
Lucifer (Greek Heosphoros-morning star, Hebrew Heylel- shining one), the son of the morning, was meant to be a type of Christ (just as all Christians are to imitate Christ), but failed because of his pride.
Now, the real issue is the meaning behind Heylel and Heosphoros. Both of which refer to the morning star, especially with it’s complimentary line “son of the morning”. The phrase that’s used to describe Jesus as morning star is completely different from the phrase that’s used to describe Satan as a morning star. And yet because these different phrases mean the same thing, this is why Lucifer was a type of Christ. Much like when Scripture says “sons of God” and “Son of God”, “spirit” and “Spirit”, “son of man” and “Son of Man”, “morning stars” and “Bright and Morning Star”. One is referring to angels or men, while the other is a divine title reserved only to God. This is why Jesus is the true Day Star.
If you're still confused over why Jesus and Lucifer are both called "morning star", then let me explain further:
The angels of God are called sons of God because they are created spirit beings meant to serve God. We Christians are called sons and daughters of God because we are sinners adopted into God’s Family through the Redemption found in Christ Jesus. Yet, there is only one true Son of God, Jesus Christ, because only he is the Eternal Son who eternally proceeds from the Father, meaning only Jesus came directly from the Father, uncreated and eternally existing with Him from all Eternity.
It’s the same then when talking about “morning star”. Lucifer is referred as a “morning star” in Isaiah 14:12. And even all the angels of God are referred to “morning stars” in Job 38:7. And yet, just like there is only one morning star (Venus) in the night sky, so there is only one true Morning Star, Jesus Christ.
While Lucifer and the angels of God are referred as “morning star(s)” in the sense that they are spirits of light reflecting the glory of God, Jesus is the Bright and Morning Star in the sense that He is the source of eternal Light and Life, for only He is the very Glory of God.
Plus, when Lucifer is referred as “morning star” the Greek word that is used is “heōsphoros” (ἑωσφόρος), which means “dawn-bringer” referring to the morning star.
However, when Jesus is referred to as the “Morning Star”, Rev. 22:16 has a far different Greek phrase used to describe Him, and that is “ho astēr ho lampros kai orthrinos”(ὁ ἀστὴρ ὁ λαμπρὸς καὶ ὀρθρινός). The Bright and Morning Star! Literally, the Star of the Bright and Morning (a descriptive phrase always only referring to Venus, the morning star). Hence, Jesus and Lucifer is not the same “morning star”, they are totally different from each other.
Let me show you the differences between Satan and Jesus by first looking at their titles each one has, and then revealing their meaning, and finally how they were translated down through time.
Let’s start with Satan in Is. 14: 21 “O Lucifer” (KJV)
Hebrew “heylel” (הֵילֵל) = shining one, boastful one
Hebrew was translated into Greek as:
Greek “heosphoros” (ἑωσφόρος) = bringer of dawn, personification of morning star
Greek was translated into Latin as:
Latin “lucifer” = shining one, morning star, bringer of light
Latin was translated to English as:
English “Lucifer” (KJV)
Greek and Hebrew was translated to English as:
English “Day Star” (ESV), “morning star” (NIV), “star of the morning” (NASB), “shining one” (NET)
Now let’s look at Jesus in 2 Pet. 1:19 “morning star”
Greek “phōsphoros” (φωσφόρος” = bearer of light, personification of morning star
Greek was translated into English as:
English “day star” (KJV), “morning star” (NKJV)
And the Greek phrase “astēr o lampros kai orthrinos” (ἀστὴρ ὁ λαμπρὸς καὶ ὀρθρινός) in Rev. 22:16 was translated as "the Bright and Morning Star"
Notice that in the original languages (Hebrew and Greek) both Satan and Jesus have very different titles. However, because of the very similar meanings of these titles, English translations have sometimes used the same words to describe both Satan and Jesus as “morning star”. For example, notice how “heosphoros” (Satan) and “phosphoros” (Jesus) are both personifications of the morning star. Yet, “heosphoros” is the bringer of dawn, while “phosphoros” is the bearer of light. Simply put, Jesus the Bearer or Source of light is superior to Lucifer who merely brings the light. Just as the angels (i.e. messengers) bring the commands and decrees of God, so Satan was once the Bringer of the light of God. But Satan was never the Bearer or Source of the light of God. Keep in mind that angels merely reflect the light of God, while Jesus Himself is the Light.
Think of it like this: Jesus is the Sun who shines on the world during the day, while the angels are the Moon reflecting/bringing the light of the Sun during the night. That is what Lucifer used to be, but not anymore.