Lyric discussion by DorianMode 

The song taunts us over and over with "What's my name?" even after the narrator tell us to just call him Lucifer.

His name is God. Not Lucifer, not Satan, but God. I used to think this verse was just bad grammar:

 As heads is tales
 Just call me Lucifer

But the anwer is right there--cop is to criminal as God is to Lucifer.

God in a bad mood is Satan. God in a bad mood lays our souls to waste:

 Just call me Lucifer,
 Cause I'm in need of some restraint.

God in a good mood plays games with us:

 What's puzzling you
 Is the nature of my game.

 What's confusing you
 Is just the nature of my game.

What a masterful song!

I always took the song at face value, but I think you just nailed it.

that is a fascinating take on it

I like this interpretation best.

Also other things that suggest it is God and not Satan:

What's puzzling you is the nature of my game. When bad things appear to happen to religious people, they say God "moves in mysterious ways" or that "we cannot understand his reasons".

And all the sinner saints, The Bible often shows a lot of sympathy for sinners, even seeming to encourage sin sometimes, check out Luke 7: 36-50. Also the Bible is often known for speaking in contradictory terms.

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