I would say it identifies the "true believers," but there's the rub. Mormons tend to be skittish to say they "believe." To some, saying "I believe the Church is true" carries with it a whiff of doubt, an element of something suspect or "less than". Sometimes a person can bear their testimony-declare their convictions--with those familiar phrases, and it is a sublime communication from spirit to spirit. Other times those phrases sound as flat as clichés or "vain repetitions." It's the Holy Spirit that communicates, but we could help It out if our language were more thoughtful and expansive.
I understand the need for using the word "know." For me the restored Gospel, the authority and ordinances of the Priesthood, the divine guidance of this Church are the truest things I know. My conviction about these things is the standard by which I measure all other things claiming to be "true" or for anything else I say I "know." To use another verb might lessen the impact, the authority, the imperative this personal revelation has for me. I know what I mean when I say I know, but I'm less certain what others mean when they say it.
Another reason I think Church members use the word "know" is to underscore the exclusivity of the truth claims of Mormondom. This is a touchy subject in our modern culture. When most people want to hear "If it works for you, then God bless," Mormons unabashedly declare that Jesus Christ is the only Way, the Truth, and the Life; that angels came again; that ancient translated books bear witness of Christ; that ordinances essential to spiritual progress are necessary for everyone and available to them now or in the eternities. Some Mormon folk may think that to say "I believe" sounds too much like the broader cultural norm of "my truth is as good/valid/right as your truth."
This last bears closer examination. Mormons (dare I say) believe that the Gospel contains all truth, wherever it is found. Brigham Young said, "If you can find a truth in heaven, earth or hell, it belongs to our doctrine. We believe it; it is ours; we claim it. Truth is all over the earth." (Discourses of Brigham Young, p. 2, 10) This means then, in fact, my truth is as good/valid/right as your truth. Identifying what is truth is a nubbier issue--the work of penitents and philosophers.
If we could shake off our Mormon cultural shivers at the distinction between "I believe" and "I know," the Church could be stronger in at least two ways. First, those who really aren't so sure about what they "know" might feel more welcomed. In Mark 9, Christ welcomed a man who acknowledges internal complexities most of us face: When asking if Christ would heal his child that man said, "Lord, I believe. Help thou my unbelief." Faith and unbelief can co-exist in the same person. People who are aware of that in themselves but are too intimidated by "Mormon-speak" to admit it, might feel valued, honored, and recognized as fully contributing members of their wards. At last.
Another benefit in broadening our rhetoric would be to acknowledge the varied ways God works with individuals. In Doctrine & Covenants 46: 13-14 the Lord says, "To some it is given by the Holy Ghost to know that Jesus Christ is the Son of God, and that he was crucified for the sins of the world. To others it is given to believe on their words, that they also might have eternal life if they continue faithful." We have "gifts differing," and it would be great to celebrate them all. Apparently God is not as hung up on words as we are.
I've had my own experience with this. Several years ago during a "season of distress and grief," I found my faith quaking. Was there even a God? Did anything I ever thought was "true" count for anything? Had I just bought into a bunch of fancy lies or made major decisions about faith and family based on emotions and brain hiccups I had only labeled as spiritual experiences? Despite all this wrestling, I knelt every night for months asking, among other things, the famous question from Moroni 10:4: Is the Book of Mormon true?
One night-and don't ask me why it happened this one night and not one of the countless others I'd been in the same posture with the same array of fragments and petitions-as I got up from my prayer, I sensed a phrase in my head. Not words I heard audibly, but words impressed on my mind:
"You're asking the wrong question. You should be asking, "Where does God want you?" I thought, What? That's not how the script is supposed to go!
As soon as I asked my new question - reframed by a Celestial Editor who knew me better than I knew myself (and Who apparently existed after all)-an answer flooded in. So here I am and here I stay, shocked and humbled. I never have received the "expected" answer to the Book of Mormon question.
I carry on in faith, a true believer.