How Evangelicals use the label "Christian" for self-definition and for certifying others
In the absence of an independent arbitrator that governs who can label themselves "Christian" (like how USDA governs the meaning of 'Organic') we will always come across different groups assigning their own precise definition, although as you say, the label should at the minimum refer to those who follow Jesus Christ, a requirement that every group at present seems to agree despite theological differences.
I think the question is best answered using the concepts of in-group vs. out-group within the discipline of the sociology of religion. For a Christian group that has a strong umbrella institution such as LDS, Roman Catholics, Church of England, etc. the "in-group" label meaning can more easily be discovered and is relatively unambiguous because the meaning is centrally defined from their official literature. But for evangelicals, they will defer to what the Bible teaches who the true followers of Jesus are as their "in-group" definition.
Of course, for the outsiders, this definition is not sufficient and is rather ambiguous, partly because evangelicals today have different stripes based on their theological persuasions:
- Some are Calvinist, some are Arminian
- Some are leaning Charismatics, some are leaning Catholics
- Some are fundamentalist, some are "big umbrella" Anglicans
Can we find a common denominator that almost all those subgroups can agree on? I think when pressed further they would say one or more of the following:
- a Christian is a follower of Jesus whom Jesus will accept on Judgment day
- a Christian is one who has the spirit of Christ living in him/her (another way of saying: born again)
- a Christian is one of the elect (whether the member of the elect is defined by Calvinism or by some other criteria)
- a Christian is one who adheres to one of the Protestant's statement of faith such as the Westminster Confession of Faith or the Thirty-nine Articles
- a Christian is one who would agree with C.S. Lewis's Mere Christianity book
- a Christian is one who "believes the gospel as taught in Scripture" defined similarly to Zanarkand's answer
- a Christian is one who holds that (Protestant) Scripture alone is the authority on doctrine, as well as sola gratia, sola fide, and solus Christus (see wberry's answer)
- a Christian is one who obeys Jesus's new commandment of loving one another and who remains within the vine (John 15:1-17)
We can easily see that if we press them further to define what it means to be accepted on Judgment day or to have the spirit of Christ living in him/her or to know whether one is a member of the elect, they will resort to their subgroup's theology. So yes, there is a theological significance to the label 'Christian' and they expect the "out-group" member to be "certified" according to their subgroup's criteria.
- the various definitions are compatible with one another, each focusing on a different aspect of "being a Christian": the intellectual aspect, the subjective spiritual awareness aspect, the ethical (doing) aspect, or the aspects that only God knows (such as "who is the elect")
- as you already observed, some "Christian" criteria ultimately include elements that only the person can know subjectively (such as being "born again")
What are some practical implications of the above approach to analyze the evangelicals's use of the term "Christian"?
This means that if an "out-group" member (such as an LDS member) wants to fellowship with an evangelical group and is being asked "are you a Christian?" this "out-group" member needs to prove to this evangelical subgroup that he/she meets the particular evangelical's "in-group" criteria to be "certified" Christian. Or to use the USDA analogy, it's like USA will insists that if a Canadian beef producer wants to label the beef they are selling in the USA as "Organic", the producer needs to go through the "USDA Organic" certification process even though it is already certified "Organic" in Canada.
Among the more generous evangelical subgroup, they will allow some out-groups members to call themselves "Christian" if they share the same definition of God and Jesus Christ. Thus Catholics are allowed to be "Christian" while still practicing Catholics if they meet their "in group" criteria, such as exhibiting a life indicating that they have the spirit of Christ living in them.
Another generous evangelical subgroup will allow some LDS members to call themselves "Christian" as long as they are willing to "walk and talk" their faith only within the subset that their "in group" allows. For example: accepting the Trinitarian definition of God, eschewing LDS-specific interpretation of the Bible, willingness to be rebaptized with the Trinitarian formula, etc.
Similarly, if an Arian / modalist shows some willingness / openness to be corrected in their understanding of God, evangelicals will also call them "Christians" if they show seriousness to be discipled in becoming true followers of Jesus according to the "in group" ethics.
The origin of Evangelicals's denying the label "Christian" to some followers of Jesus
The previous section shows how Evangelicals tend to associate the label "Christian" with theologically laden definitions, which then serves as a standard for the "purity of the brand", which they then apply to judge their own churches/members (see wberry's answer) as well as to judge non-Evangelical churches/members.
What significance has the label Christian that causes a reluctance to grant it? Are all Christians saved? Have Christians no need of being the target of missionary work? Do they not need to be rebaptized if they already are baptized in a different denomination? The answer to all these seems no, or at least "it depends". I have a reasonably good idea what offends evangelicals about LDS doctrine, or catholic doctrine, but why does this lead to "you're not Christian"? What is the gain of denying the label?
I believe looking at the history of the evangelical movement is instructive here, especially the early 20th century wave: the Fundamentalist movement. Fundamentalists started as a reaction against liberal leaning Christian theologians who watered down the supernatural nature of the Bible. To them being a 'Fundamentalist' was a badge of honour (unlike today's perjorative connotation). Evangelicals see themselves as the more enlightened generation of fundamentalists but still share many of the same principles. To distinguish themselves from other Christians they call themselves 'evangelical' or 'born again Christian'. Now, 100 years later, especially after the Catholic church see Protestants as "separated brethren" (rather than "heretics" before VC II), evangelicals became more comfortable to reach out to Catholics and Orthodox to affirm what unite them rather than what divide them, so the label "Christian" started to be used for this larger "in-group" identity.
You asked: "What is the gain of denying non evangelicals the 'Christian' label?" My answer is: like their fundamentalist forbear, evangelicals have the habit to use a label as a moniker for doctrinal purity. Old habits die hard. It's a defensive mechanism that was critical 100 years ago although not so much now. Among some evangelicals the label 'Christian' is now regularly used for anyone affirming the Nicene or at least the Apostles' creed, not restricted any longer to those who hold the sola scriptura position. As long as they still see LDS, JW, Unitarian, and other groups as a threat to their doctrinal purity, I believe evangelicals will continue using the label 'Christian' as an in-group identity rather than to label those who are truly saved in the eyes of God. (Of course there are exceptions since evangelicals are so broad nowadays).
In contrast, the Catholic Church is now one of the most progressive among major Christian denominations to recognize the possibility that an adherent of other religions may access salvation (which is in Christ alone), a position defended in the 2017 Angelicum journal article The Salvation of non-Christians? Reflections on Vatican II’s Gaudium et Spes 22, Lumen Gentium 16, Gerald O’Collins, S.J. and St. John Paul II:
... I defend the Church’s teaching that Jesus Christ’s redemptive work is the full and sufficient means of salvation, and hence that the mission of the Church is to call all men to Jesus Christ who is the only mediator between God and man (1 Tim 2:5). Regarding the fate of the unevangelized, namely, those who through no fault of their own—the invincibly ignorant—have failed to respond to the Gospel, I argue for accessibilism and rejected inclusivism. I also underscore the importance of the distinction between the sufficiency and efficacy of Christ’s atoning work, and in this connection the distinction objective and subjective salvation.
In John Paul II's encyclical, Redemptor Hominis, he underscores the actual impact of the Incarnation has given that the Son of God has united himself, in a certain way, with every human being. What does John Paul mean when he states that all men, without exception, from the very first moment of their existence, in short, from their conception, begin to share and are included in the "mystery of redemption" by virtue of the Word's mode of union for each person in Christ?¹³⁶
... my thesis is -- following Caroline Farey -- that there are "progressive stages of increasing configuration in and to Christ. The union of Christ 'with each man' at the Incarnation, then, is totally for this kind of increasing union, in the Church. From the historic moment of the Incarnation in the womb of Mary Christ begins 'to reconcile to himself all things' [Col 1:20]."¹³⁷
but the author carefully distinguishes his position (called accessibilism) from inclusivism. It is then natural that Roman Catholics, who already have a label for self-identity ("Catholic"), will then more likely (in the future) to use the label 'Christian' for anyone who responds to the inner call of Christ regardless of their present religious affiliation. I think this started with Karl Rahner's Anonymous Christian notion.
Evangelicals are not there yet, maybe never. At present they don't even consider all Catholics who die in the state of grace are necessarily saved even though they have been baptized in the same Trinitarian formula that evangelicals use.
As a group which grew out of the Protestant tradition, Evangelicals tend to label a whole "out group" to be "not Christian" if the out-group's statement of faith is deemed not to be compatible with Protestantism (such as LDS, JW, Arian, Modalists, Unitarian, etc.). Since Catholicism and various Orthodox denominations share the same ecumenical creed(s), Evangelicals tend to label them "Christian" too, although with some reservations (treating them not wholly "in-group" but not wholly "out-group" either). However, Evangelicals do make exceptions for individual members of the out-group if they are willing to be "certified" Christian using the in-group's criteria. The "in-group" criteria itself is stable and theologically well defined enough within the subgroup (despite various ways to define it) for anyone to use for "self-certification".