We must think of these characters in the context of The Fall - the War in Heaven. Many traditions use the names interchangeably to represent the Devil. In most traditions, the Devil is disparaged as the true evil, while in many Lucifer is regarded as the Bringer of Light - the model for Prometheus. Steiner for example, though I think Lucifer is less than perfect in his model
Christians are most likely to see the two as interchangeable, while Blavatski’s Theosophy (the foundation stone for the New Age Movement) will hail the Light Bringer. This is a broad and deep rabbit hole, and your question, by itself, has the power to unravel everything you understand about the world you live in. So grab a shovel if you dare, pick a depth, and jump in.
I would suggest this entry point, while others may vehemently disagree (according to doctrine): Think of Lucifer as the grand architect of the rebellion against God. Think of the devil (Satan) as the lord or prince of this world. The two are/were collaborators. This war in heaven led to the ‘fall of man’ according to the Abrahamic religions, with Satan the tempter of Eve. And now you are in the thick of a battle. The rulers of this world support the Light of Lucifer in many guises, including secret societies, and even cripple Christianity with the incursions of gnostic thought and new-age sensibilities.
Have fun.
Alan