Here is my understanding.
Satan (from a Hebrew word meaning “adversary”) is the name we give to the fallen angel who “rules” (for want of a better term) over Hell. There may be other devils (fallen angels), but Satan is “the” Devil. He appeared in the garden of Eden and tempted Adam and Eve, made life difficult for Job, tempted Jesus in the desert, etc etc.
Lucifer (Latin, “light-bearer”) as applied to Satan comes from the book of Daniel, and some hold the passage in question to be actually addressed directly to Nebuchadnezzar, and obliquely to Satan. In the Vulgate, the word *lucifer * is applied also to Jesus (as a bringer of light) but in a different context. There was a bishop Lucifer back in the days of the early church, and I believe in one of the eastern churches he is venerated as a saint.
So there’s the Devil – a.k.a. Satan – a fallen angel whom we reject with “all his works and pomps”, who is in charge of a bunch of other devils, or demons, or evil/unclean spirits, or fallen angels.
Now as for all this crown prince of hell business – this sounds like a lot of medieval demonology which tried to describe some sort of infernal hierarchy, matching seven devils with the seven cardinal sins, deciding that Hell had nine circles*, figuring which devil originally came from which choir of angels (was Satan originally a cherub or a seraph?), yadda yadda yadda.
Basically, The Devil = Satan, he’s bad news, avoid him. “Lucifer” may be an oblique reference to him. But they are not three “separate” individual angels. I do not believe there is any real support either in the Bible, Tradition, or any Church teachings.
My recommendation is to avoid demonology (and angelology, while you’re at it).
*And since one of those circles is supposed to be a frozen wasteland, it means that a snowball’s chance in hell is actually 1 in 9, or a little over 11%.