Something else I'll leave is an explanation from another regarding Satan and Lucifer.
Many believe that they are one and the same. The idea is that 'Lucifer' which translates as 'light bearer' or 'morning star' was the name of the Angelic being who became Satan, before he fell from grace. ('Satan' itself means 'the deceiver').
There is a tradition that he was originally an archangel who fell due to pride. Many in the church beleive this.
Curiously there is nowhere in the Bible that definitely states that Lucifer and Satan are the same entity, or that he was an archangel. The only archangel named in the Bible is Michael. (There are others mentioned in the apocryphal books.)
Since God is not evil and cannot create anything evil, it follows that the devil probably did start out as an angelic being. As to what kind, we can't know for sure, but he was likely high ranking. Quite possibly he was an archangel or a cherub, but the Bible does not state for sure. It is clear however that he rebelled and took a large portion of the Angelic host along with him. The book of Revelation is quite clear on this:
And there was war in heaven: Michael and his angels fought against the dragon; and the dragon fought and his angels, And prevailed not; neither was their place found any more in heaven. And the great dragon was cast out, that old serpent, called the Devil, and Satan, which deceiveth the whole world: he was cast out into the earth, and his angels were cast out with him.
Revelation 12:7-9 KJV
As for 'Lucifer’ we see that word used in the following passage:
That thou shalt take up this proverb against the king of Babylon, and say, How hath the oppressor ceased! the golden city ceased!
Isaiah 14:4. KJV (I include this for context as it is the start of the proverb)
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! 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. Yet thou shalt be brought down to hell, to the sides of the pit.
Isaiah 12-15 KJV
It's clear from verses 12–15 as to why many believe this 'Lucifer' is in fact Satan since the tale of Lucifer's pride and attempt to ascend to the throne of God certainly coresponds with what we read concerning Satan in Revelation.
That being said, while I think this may be a valid interpretation it is not definitive. Nowhere does it state that these two entities are the same. In fact verse 4 clearly indicates that this proverb is aimed at the King of Babylon, who had the people of Israel in captivity at this time. The verse that succeeds this passage also indicates a human being:
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;
Isaiah 14:16 KJV
Some would argue verses 12–15 is aimed at the spiritual principality behind the king of Babylon, which again may be a valid interpretation. Or the entire proverb could be a disclaimer against a human king using metaphorical language. Or it could be aimed at both the man and the spiritual entity behind him.
That all being said, there is nothing in these verses that unequivocally states that this 'Lucifer’ is the devil spoken of in Revelation, although I think it is very possible.
So, Satan (meaning accuser) may also be a mantle that Lucifer was given after his fall from Heaven. If we look at it this way, after the final battle, Lucifer is Satan and cast into the Lake of Fire for eternity, then Jesus takes on the role of Morning Star in the New Heaven. So, NO, Jesus is not the Fallen Angel Lucifer.