It depends which book you’re reading.
According to the N.T and Quran. Satan is the first Fallen Angel. But Satan is also a title or a name for a race of spirits, the Fallen Angels or Demons. All those spirits can act or manifest as one being. The word “Satan” has different meanings. One of them, is the well known “adversary” that comes from the Hebrew “Ha Satan” . The other, is from the root “Shait” which can be found in how the name is pronounced in Arabic for example. Which is pronounced “Shaitan”. The meaning of this word describes the nature of a race of spirits, also known as Jinn in Islam and specific type of Jinn, that we call “Demons”. In Islam, Not all Jinn are Demons but all Demons are Jinn.
Therefore, Satan is a name of the first Fallen Angel. A title for a race of spirits, Demons. And he’s also “The All” or the God spirit for that race.
According to the O.T, Ha-Satan, is a job title for anyone that God uses to test the faith of believers. Could be an Angel or a human being or anything else. In the O.T you’ll find Angels called “Satan”, and interestingly in the N.T Jesus called saint Peter “Satan”. Because both of them were Jewish and understood what that word means. When Christianity was invented or developed as a separated religion, they couldn’t take that part away because it predated the religion itself. So it’s still there. But almost no one speaks of the meaning of it because it would cause huge problems for the Church.
As a “Fallen Angel” the name “Lucifer” was given to Satan by Christianity. But it really doesn’t exist in the bible. The word “Lucifer” is mentioned in the bible twice, one time to describe the king of Babylon. And the other time to describe Jesus. In both cases, the one called “Lucifer” is the planet Venus. Which can be seen in the sky twice a day. At sunset, it’s called the “evening star” and that manifestation was used to describe the king of Babylon. Because it’s a bad sign that predicts the coming of darkness. The second time Venus can be seen, is at dawn. And considered a good sign because it predicts the coming of light or sunrise. That manifestation is called “the morning star” and was used to describe Jesus. Lucifer was not used as a name of a being or a spirit or a devil, anywhere in the bible. That meaning was a mix of wrong translations, misunderstandings and adaptation of Greek mythology. In other words, Lucifer is only Satan according to a Christian mistake. Lucifer is not Satan, got nothing to do with anything in the bible or Christianity, including Demons!
People can believe what they want about anything or anyone, that’s a personal free choice. But when it comes to asking about the origin of a character that belongs to any culture or religion. The sources speak one language and tell a specific story. Anything outside the range of those sources, is a personal belief or opinion. Nobody could ever say “Zeus for me is such and such” because we can get back to Greek mythology and learn who he is. The same with Satan and Lucifer or any other spiritual being. I can’t tell you “Satan for me is this and that” I can only tell you which book or religion I believe in, about Satan, and my experience in working with him. Which should not contradict his character in that source or something would be wrong with my head or those sources need to be proven inaccurate.