No, it doesn't say that believing means overcoming. The one who overcomes does so because of his faith, which the Bible tells us as what actually overcomes the world. 1 Jn 5:4 says that.
"For whatever is born of God overcomes the world; and this is the victory that has overcome the world—our faith."
What is "our faith"? Because it is a NOUN, and NOT an action, it refers to what we believe. Which is where all power is.
You're trying to make a noun act like a verb. But it isn't and doesn't.
No faith - No overcome
No overcome - No book of life.
Question: Who is he that overcomes?
Answer: he who believes...
4 For whatever is born of God overcomes the world. And this is the victory that has overcome the world—our faith.
5 Who is he who overcomes the world, but he who believes that Jesus is the Son of God? 1 John 5:4-5
Those who believe for a while, are saved for a while.
Those who believe for a while, overcome for a while.
Those who endure to the end will be saved. Matthew 24:13
... be faithful unto death, and I will give you the crown of life. Revelation 2:10
He who overcomes shall be clothed in white garments, and I will not blot out his name from the Book of Life; Revelation 3:5
“Who are these arrayed in white robes, and where did they come from?” And I said to him, “Sir, you know.”
So he said to me, “These are the ones who come out of the great tribulation, and washed their robes and made them white in the blood of the Lamb. Revelation 7:13-14
JLB