Christianity is both personal and communal, and ultimately traces historically back to the nation of Israel, the person of Jesus, and the teachings of the Apostles. Someone claiming Christianity as a personal identity which does not correlate to the communal and historical understandings of Christianity is, at best, using a label in a misleading fashion. (They may be lying, they may be confused, they may genuinely disagree, but ultimately they're using the word in a confusing way -- claiming membership in a community and participation in a history which they reject.)

The "big three" splits in Christianity can all be traced to genuine disagreements while retaining major agreement over key points. Orthodoxy and Catholicism split over the authority of the Roman leader (the Pope) as either equal to the leaders of the other main centers of Christianity or as supreme over them. Catholicism and Protestantism split over the authority of the Bible relative to the authority of tradition and contemporary church leaders. Various denominations within Protestantism split from each other over disagreements ranging from how baptism should be performed (dunking vs sprinkling) to whether women can be ordained as pastors. But through it all, there is a broad set of agreement -- that Jesus is God (connected to Trinitarian Monotheism), that the Bible is foundational for Christian belief, that fellowship with other believers helps refine and center Christians (consider phrases like "the body of Christ" and repeated references to "The Church" within the New Testament), and the centrality of grace/repentance/forgiveness to life as a Christian. While uneducated laymen from these groups might not recognize one another, among educated laymen, scholars, and theologians, there's like 99% agreement that they're all the same basic religion.

There are fringe groups that don't follow this pattern. Mormonism is one of them -- at its founding, its "prophet", Joseph Smith, claimed that all of Christianity was an abomination, and that he alone held the keys to "Restored Christianity". He introduced three new foundational books which are treated with higher reverence than the Bible (the Book of Mormon, Doctrine & Covenants, and Pearl of Great Price). These books introduce "alternative history" -- the new people, events, cities, and technology found in the BoM have no historical or archaeological grounding. And these books (mostly D&C) introduce new beliefs and doctrines that are not found anywhere within Christian history, not in either mainstream or fringe groups. As such, Mormonism is rightly considered to be a new religion. (As an interesting aside, most Mormons did not consider themselves to be Christian up until the last few decades.) Likewise, Joel Osteen is broadly considered to not be teaching Christianity -- he might personally be a Christian, but he doesn't teach the stuff that every major branch of Christianity agrees is critical for being a Christian, like the need for repentance and forgiveness. The same can be said of various other fringe groups, televised preachers, etc.

This matches up pretty well with the above comment about self-identifying as a scientist. The word actually has meaning beyond personal identity. It implies things about your community and about your beliefs/approach to knowledge meeting the standards of that community. And while there are always people on the fringes who claim the label, and always gray areas, you can get a pretty good read by going to a known-good source and then building a sort of network of trust from there.