The Book of Life is just that - a book of every person ever born. The only names that get removed are those that have ultimately rejected salvation and are destined for spiritual death no longer candidates for eternal life.
I agree . . everyone's name is in the Book of Life ~~ and to those who reject the Gift of salvation and reject God's Son is blotted out
It is my understanding that there are two books: The Lamb's Book of Life, where a believer's name is written in when they accept Jesus as Lord and Savior, and there is also a "book of the living" as King David wrote about in Psalm 69:27-29:
Add iniquity to their iniquity,
And let them not come into Your righteousness.
Let them be blotted out of the book of the living,
And not be written with the righteous.
I get the impression from this verse that everyone's name is initially written in the book of the living, but their name is blotted out when they they refuse to repent from their unrighteousness. Then this verse goes on and refers to their name not being written with the righteous, which seems to me that is talking about the Lamb's Book of Life.
So when their name is blotted out, it's blotted out of the book of the living, not the Lamb's Book of Life. I think that's where the term being "blotted out" comes from.
That's what I was taught, anyway.