Both Revelation 20:14 and Rev 21:8 refer to the second death of the ungodly.
But those who do believe and keep their faith thru all the testing and trials to come, are blessed and holy and over them the second death has no power. Revelation 29:6
Their names are found in the Book of Life. Malachi 3:16
The second death happens after the Great White Throne Judgment, where everyone who has ever lived, are brought back to consciousness and will stand before God. Revelation 29:11-14, Daniel 7:9-10 Their names are not found in the Book of Life.
1 Corinthians 15:50-56 describes that time and the interesting fact that the faithful people who are alive then, will never die, but be changed directly into immortality. Revelation 21:1-7
The living ungodly, like all that army described in Rev 20:7-10; including Satan and his fallen angels, will die and never live again.
Whether the Lake of Fire for them, lasts for Eternity or not does not concern us.
That 1 Corinthians quote, is the main one touted by the ‘rapture to heaven’ believers. I am vilified and abused when I point out that it happens at the GWT, as only then is Death no more.
We are plainly told that people will die during the Millennium. Isaiah 65:20