The question here hinges on whose definition of "Christianity" is used. We could use a Reformed definition, an Arminian definition, a Roman Catholic definition, an Eastern Orthodox definition... or maybe an Anglican, or Episcopalian definition... how about the Pentecostals? The criteria for those are described fairly clearly. We could also use definitions that are less specific: how about an Islamist definition, a secular humanist definition, or a "new atheist" definition?
Edit: The difficulty with finding "the bible's definition" of who is/isn't a Christian, or of nearly anything else, is that -- even for those who believe that it has only one Author -- the bible remains is a massive book, written over the course of several centuries, in multiple languages, by many different people, living in significantly different cultures. It requires intensive, thorough interpretation. For any one definition that is offered, several more, often mutually exclusive definitions can be immediately provided -- even when all were develop by teachers that were among the most brilliant bible scholars in all of history. Pick an obscure -- or not-so-obscure -- theological concept, and look up how it was taught by, for example, Aquinas, and Calvin, and Wesley. It can get rather complicated.
In any event, the word "Christians" refers simply to a group of people, but of those who self-identify as Christian, I anticipate that the majority would offer a more specific classification of themselves: Baptist, Methodist, Roman Catholic, Presbyterian, "Assembly of God-ite" (that's really what some AoG adherents call themselves, in a friendly, tongue-in-cheek way), etc. And each of those groups has specific criteria that can be used to determine who is, and who isn't, a member of their community. Most (all?) denominations have a formal "Statement of Faith" describing their doctrinal stance, and many churches require that any individual who wishes to join their membership first complete an extended Sunday-school type course that includes basic doctrinal instruction, perhaps over a weekend. Some also require the study of a catechism.
The centrality of the bible results from the fact that it is widely regarded as the official primary source document of Christianity.
I am aware of no groups that officially use the word "Christian," without qualifiers, to describe themselves. In practice, however, we unfortunately find that there are many who are anxious to claim exclusive ownership of the term, and denigrate any and all who disagree with their particular interpretive stance.
Ultimately, if the bible is true, it is the God of the bible who will judge whether or not any particular individual is, or is not, a "Christian."