Do LDS believers not accept any others who self-identify as 'Christian'?

Jesus Christ Himself doesn't agree with your premise

Not every one that saith unto me, Lord, Lord, shall enter into the kingdom of heaven; but he that doeth the will of my Father which is in heaven. Many will say to me in that day, Lord, Lord, have we not prophesied in thy name? and in thy name have cast out devils? and in thy name done many wonderful works? And then will I profess unto them, I never knew you: depart from me, ye that work iniquity. from me, ye that work iniquity. (Matthew 7:21-23)

The Lord taught that some people who self-identify as Christian will be excluded from His kingdom. But He also taught something else that the LDS Church completely believes in:

And John answered and said, Master, we saw one casting out devils in thy name; and we forbad him, because he followeth not with us. And Jesus said unto him, Forbid him not: for he that is not against us is for us. (Luke 9:49-50)

Those two verses seem to identify a paradox. On the one hand, people who self-identify as Christian are excluded. On the other hand, people who self-identify as Christian are included.

What the LDS Church believes is that Jesus' Kingdom is divided into various glories.¹ The highest glory is reserved for those who embrace the testimony of Christ fully and who receive covenants and ordinances via the full authority of Christ (D&C 76:50-70).

Others receive a lesser glory, depending on their choices and commitment to the fullness of the gospel of Jesus Christ (D&C 76:71-89).²

So, where does the LDS Church draw the official line? Let's look at Alma 5:60:

And now I say unto you that the good shepherd doth call after you; and if you will hearken unto his voice he will bring you into his fold, and ye are his sheep; and he commandeth you that ye suffer no ravenous wolf to enter among you, that ye may not be destroyed.

So the line is drawn at people who would destroy the Church. Whether they self-identify as Christian or not. This sets the limit for exclusivity — but what about inclusivity? Let's look at D&C 46:3-5:

Nevertheless ye are commanded never to cast any one out from your public meetings, which are held before the world. Ye are also commanded not to cast any one who belongeth to the church out of your sacrament meetings; nevertheless, if any have trespassed, let him not partake until he makes reconciliation. And again I say unto you, ye shall not cast any out of your sacrament meetings who are earnestly seeking the kingdom—I speak this concerning those who are not of the church.

We are commanded of the Lord to not exclude anyone from our public meetings. This reflects Luke 9:50.

But what did you actually mean?

Unfortunately, your question is unclear. What do you mean by "reject" and "disallow?" There's a difference between how individual members of the Church act and what the Church teaches, depending on what you actually mean by those words.

It is an unfortunate and painful truth that individual members can and do act with introversion and exclusion. This is not the gospel of Jesus Christ, but like the ancient Jews of old, being told that:

• Yours is the "only true church of Jesus Christ."
• You are the "elect of God."
• You are the "chosen people."

can go to your head. It is a sign of apostasy to exclude others when the goal is to embrace them in love and offer them (whether they choose to accept it or not) every blessing the Lord has to offer. It is a constant struggle to help members avoid this particular sin.

But if what you meant by your question was something along the lines of what @DJClayworth mentioned, that "accept" and "allow" means to consider the authority, gospel, and opportunities offered by another church as equal to our own. The answer is no. We believe that no other church offers the fullness of the gospel of Jesus Christ, the fullness of the authority of the Melchizedek Priesthood, and therefore has not the authority to open the door to Christ's exaltation.

We accept all who are not against us as Christians, but we recognize (as the Lord taught) that not all will embrace the highest glory He has offered in His kingdom. And that includes our own membership.

_{¹ Detailed explanations of the Degrees of Glory have been given elsewhere on this Stack. Rather than clutter this question with yet another detailed explanation, I'll summarize and otherwise defer to them.}

_{² I'm not summarizing Outer Darkness or Perdition here. It's a kingdom without glory, though still within the judgement and therefore kingdom of Christ. But it's not relevant to the discussion concerning those who self-identify as Christian unless they also happen to be someone entirely against Christ in all forms and in all ways. Bit of an oxymoron, there, but I guess it could occur.}

the only true and living church upon the face of the whole earth¹

Says something about organizations, and does not discount anyone as "not Christian". A Christian is someone who follows Jesus Christ.

only: there is no other organization in which the authority, and the revelation, is found. This is not a judgement of individual people, or even organizations, that is just what God told Joseph Smith in the First Vision, long before any call to restore the church.

true: one one hand, true doctrine, but more importantly, truly the literal church of Jesus Christ, with his authority to baptize etc. This does not mean that everything that is true is already known.

living: still led through continuous revelation. This is important because even if the doctrine was true, but no further communication from God was received, the true doctrine doesn't really help because what you really need is to know what God wants right now. Also, eventually errors are bound to creep in if there is no constant guidance.

If you take the church today, and copy-paste it, then you would of course have a hard time to discern the two at first on a doctrinal level, but one would still be the only true and living church and the other one would just be one of the other ones that aren't.

_{1 D&C 1:30}

_{See also The Only True and Living Church or The Only True and Living Church (they are different articles by different LDS leaders)}

The LDS have a slightly differing definition of Christian:

Though the label “Christian” is often associated with particular creedal claims which the Church does not adopt, Latter-day Saints use it to express their belief in the divinity of Jesus Christ.

Confusion sometimes arises over whether Latter-day Saints are Christians due to doctrinal differences from those beliefs that have typically defined many other Christian churches. For example, Latter-day Saints believe that God the Father loves, weeps and rejoices with a sympathetic heart (Moses 7:28-33). They believe that the Trinity (God the Father, Christ the Son and the Holy Ghost) are three distinct personages (Doctrine and Covenants 130:22). Latter-day Saints also believe that Christ established a church during His mortal ministry, but over time its line of priestly authority, as well as certain important teachings, were lost. For this reason, they believe God called a prophet, Joseph Smith, to reestablish Christ’s church of ancient times — hence the name, The Church of Jesus Christ of Latter-day Saints (or, followers in these present times). This included reinstating living “apostles and prophets” (Ephesians 2:20). Their belief in an open canon and additional scripture — the Book of Mormon, the Doctrine and Covenants and the Pearl of Great Price — also set them apart from other Christians.¹

The LDS believe any who follows Jesus Christ are Christian.²

While members of The Church of Jesus Christ of Latter-day Saints have no desire to compromise the distinctiveness of the restored Church of Jesus Christ, they wish to work together with other Christians—and people of all faiths—to recognize and remedy many of the moral and family issues faced by society. The Christian conversation is richer for what the Latter-day Saints bring to the table. There is no good reason for Christian faiths to ostracize each other when there has never been more urgent need for unity in proclaiming the divinity and teachings of Jesus Christ.¹

_{1 Are Latter-day Saints Christian? and Are Mormons Christian?}

_{2 Are Mormons Christian? (different article)}

_{See also Roman Catholicism-LDS article, references other denominations as being Christian}

When most Christians say they "accept another denomination as Christian" it means that they recognize them as genuine followers of Jesus, saved and redeemed, part of the true Body of Christ. It's a statement that the other denomination is equal to them before Christ. They may still believe that their denomination has "more" truth, but the differences are recognized to be minor, or at least acceptable.

The Latter Day Saints treat this differently. They see 2 levels:

1. Latter Day Saints, the true church of Christ, the only one with the authority and necessary teaching to be true followers of Jesus and to be 'saved'.
2. "Christians", which includes those inside and outside the LDS church. Except LDS they have no priesthood authority¹, have incorrect and insufficient teaching, but are doing their best to be followers of Jesus under the circumstances.²

So when Latter Day Saints say they "accept someone as Christian" it means they fit firmly into that second category - believers who don't have the authority or truth that the LDS have.

This explains why LDS can call someone "Christian" but also feel the need to evangelize them, and if possible convert them to the "full" "correct" Latter Day Saints belief.

This two level belief explains the contradiction between the LDS being

the only true and living church on the face of the earth . . . .

while still calling other believers "Christian".

So while Latter Day Saints can say "we accept others as Christian", by their own definition of Christian, they do not accept other denominations as equals of them in the way that most Christians mean it.

Sources: Joseph F. Smith, ed., Teachings of the Prophet Joseph Smith [TPJSI, Salt Lake City: Desert Book, 1976, p. 372 (cf. D&C 20:38-58, 107:13.14.20) How does the view of the Church of Jesus Christ of Latter Day Saints about Jesus Christ differ from Protestant Trinitarian Christology?
as per answer given by SLM

_{1 Priesthood and Articles of Faith, James E Talmage}

_{2 Joseph Smith First Vision Account and God Love's Mankind (starting at 15 February 1978)}