What biblical reference is there that uses the word "believer" and says that person isn't saved?
The term "believer" is frequently used to refer to those had entered the kingdom of the Lord. For example, "And believers were increasingly added to the Lord, multitudes of both men and women" (Acts 5:14). (See also Acts 10:45; 16:1; II Corinthians 6:15; I Thessalonians 1:7; 2:10; I Timothy 4:10; 5:16; 6:2; I Peter 1:21).
However, every person who believes is not necessarily a believer in this biblical sense. James tells us, "You believe that there is one God. You do well. Even the demons believe--and tremble!" (James 2:19). Demons believe in God, but their belief does not indicate that they are saved. You could rightly say demons believe, but they are not believers.
Cornelius was a man of faith (Acts 10:2), but he was instructed by God to send for Peter, "who will tell you words by which you and all your household will be saved" (Acts 11:14). Cornelius believed, but he did not become a believer until he was added to the Lord. "And with many other words he testified and exhorted them, saying, "Be saved from this perverse generation." Then those who gladly received his word were baptized; and that day about three thousand souls were added to them. ... And the Lord added to the church daily those who were being saved." (Acts 2:40-41, 47). Cornelius did the same as those found in Acts 2. He and his household were baptized (Acts 10:47-48); hence, they were added to the church and thereby became numbered among the believers.