Is Every True Believer a Disciple?

There are some professing Christians who claim that there is a distinction between a Christian believer and a disciple. For instance, consider the four differences between 'believer' and 'disciple' proposed here. Or, the seven differences listed in Dr. Rick Flanders' articles here and here. Or Bob Wilkin's primary distinction of being under Biblical instruction, here. According to this form of doctrine, there can be disciples who are not believers, believers who are not disciples, and disciples who are believers. You can find many other web sources that teach this doctrine.

Other online sources oppose this teaching. For instance, here.

As I read through John 18 this morning, I wondered if Peter would have consoled himself with that kind of doctrine after he denied Christ. Remember that in John 18:17, a servant girl asked him if he was a disciple of Jesus, and he denied it. He denied being Jesus' disciple again in John 18:25-27. Do you expect that Peter said to himself, "Self, I just denied being a disciple of Jesus, but at least I did not deny being a believer in Jesus"? Such a thought seems impossible. Peter denied being a disciple and a believer in his "I am not" statement. For those who hold to the "believer distinct from disciple" theology, and who consider themselves believers but not disciples, I would simply ask, do you think it is realistic to try to affirm that you are a believer but not a disciple? How do you differentiate your denial of being a disciple from Peter's denial of being a disciple?