There's also the fact Satan and Lucifer were never the same entity.
Benefits of playing SMT - you learn this.
Satan was basically God's prosecuting attorney - God presented the arguments that the individual was good, Satan presented the argument the individual was bad, and did stuff like the Book of Job and other such things to test Man's piety; if they failed, Satan used that against them in court so to speak (this is as abjectly unfair as it sounds). Satan was the "Prince of Darkness" in a different way than how Lucifer is often presented. Basically from all the early sources he was the dirty jobs guy on top of the other stuff. This comes into play in SMT in that Satan NEVER outright presents himself AS Satan. He's always subdivided up, hidden, or out of plain sight.
There's also the...odd...Angel Samael. No two accounts on this serpentine customer can agree if he's Fallen or in YHVH's good graces. He shows up in a vision of Heaven Moses has...and causes the poor guy to flip out. His Angel guide then says basically "don't you worry - you won't EVER have to deal with this fucker." (Going by the art I do not blame Moses for his reaction
https://vignette.wikia.nocookie.net/megamitensei/images/5/59/Samael.jpg/revision/latest?cb=20110611102937). Reason I bring him up is because he's often conflated with the snake of Eden, associated with Satan...and occasionally Lucifer, fueling the "whose side is this guy on anyways?" theological fun.
Lucifer is more like a celestial gadfly. He's always up to no good - with his more prominent position largely springing from the book "Paradise Lost" by one John Milton and then exploding from there. Even prior, Early Church leaders knew they needed to make evil diabolical but also seductive to try and prop up their side as more righteous - so Lucifer became a cunning, plotting figure. His characterization as a sort of "Demon Lord" also likely springs from Milton, where he commands the likes of Beelzebub, Belial, Lucifuge, and other assorted figures. (note from this that could be somewhat amusing; Hell has apparently had multiple "Presidents". From all I gathered, it seems they have a split executive branch where Lucifer is a Prime Minister / President for Life, with various Presidents such as Ose, Lucifuge, and others. Y'know, kind of like Russia, where Putin seems to be in charge forever and rotating other people around as one position or the other.)
As for why I bring this up, well, I figured this was as good a thread as any for a bit of a myth history lesson, and I do adore mythology. And I figured this all might just melt Jacob's brain down. Let's see him try to turn the Satan =/= Lucifer into a whole massive argument...