What does the statement "The (LDS) Church is true" even mean...?

“The Church is true” is a statement I learned to make as a child; I learned to repeat the phrase mechanically before I had the cognitive wherewithal to evaluate the meaning of it. It is not a sentence about which I can say I honestly know what it means; it seems to me now to be essentially meaningless because it is, I think, a categorical error.

If I say “my daughter came home in a flood of tears” and you respond “really? I thought she came home on a train…” you have somehow missed the point of what I was trying to convey. Your error ought to be obvious. You treated the flood of tears and the train as though they are both the same sorts of things—as if they belong in the same category. Even though the train and the flood of tears might structurally fit into a sentence in exactly the same place, they change the meaning of the sentence by virtue of the fact that the flood of tears belongs to the category of emotional states, whereas the train belongs to the category of means of transportation.
A categorical error occurs when we conflate things that belong in one category with things that belong in another category. It was introduced by Gilbert Ryle (1949, The Concept of Mind. University of Chicago Press) as a way of resolving the issue of the relationship between mind and body. He suggested, contra Descartes, that mind and body belong to different categories.
There is a category of things that can potentially be true or false. 
Ask yourself: What sorts of things can be true and false? Is a question true/false? Or a rock? Can a rock be true or false?
That category can be described as things that are representations of other things. Representations are things that are about something (usually something other than itself). Photographs, for example, are representations, because they are about something (a family, a party, a UFO…), books or films are about something. A memory, a belief or a proposition are all about something. 
Forgive the impreciseness of my language here, but here is a working definition of a representation: A representation is something that is about something.
Things in this category (things that are about something) might be accurate representations or they might not be. A book about WWII might contain errors. It is still a representation, just not an accurate one. A memory or belief may or may not accurately coincide with the way things actually were or are.
So this category of things can be true or they can be false, or can be a mix. Your memory of your 10th birthday party probably contains parts that are true and some that are not. For the purposes of this essay we will hold that a representation is true to the extent that it accurately corresponds to the facts—the way things are (or were, or will be).
So, quite simply, if I believe that Mt. Kilimanjaro is the tallest mountain in the world, is my belief true? It (my belief) is true if and only if, in reality Mt. Kilimanjaro is the tallest mountain in the world. If a sentence states the proposition “God exists,” then that sentence is true if and only if there is a fact that corresponds with that proposition. Conversely, if a photograph, memory, dream, etc. represents something in a way that does not correspond with the way things really are, then we can say that that photograph, memory, dream, etc is not true. It is not a true representation of the facts.
With that definition of the category of things that can be true in hand, it is now easier to see why “The Church is true” is a problematic statement.
There are clearly things that don’t fall into the category of things that can be true. Unlike propositions, some statements are not representations—questions, or performative utterances (“I do,” “I promise.”) do not represent the way things are. People are not representations (…well, maybe actors…), and so are not true in the sense that representations can be true.
Institutions are not representations. Would it make sense to say “your family is true” or that “the Provincial Government of Sakatchewan is true” or that “the Prime Minister of Great Britain is true?” Not in the same sense that a history book or a theory could be described as true.
Would it make sense to say “my family is false? No? Why not? Because families are not representations, and do not fall under the category of things that are about something, so cannot be true or false. Similarly, the LDS church, being an institution, is not about something else. It does not purport to represent, in the same way that a picture, memory, painting, sentence, or belief does, something else. It lacks aboutness. So the LDS church, like any other church, or any other institution, does not fall into the same category as those things that we might describe as being true or false.
Yet we still hear the phrase (“the Church is true”). And frequently. So what do we mean when we utter such a phrase. I suspect that in each speakers mind the sentence might mean something slightly different, but I can think of a few plausible interpretations of what it might mean to an individual.
“The LDS church is the only one with the real Priesthood, or with legitimate God given authority.”
“The LDS church is the only one led by a divinely inspired prophet, the only one with God at the helm.”
“The sum total of the teachings of the LDS church are true.”
“The core doctrines of the LDS church are true.”
“The LDS church is necessary for salvation; the LDS church is the only one that can offer salvation.”
I suppose we could extend this list further, but the list is meant to be representative, not exhaustive. In each case, it is not the institution itself that is true or false, but some statement or belief about the Church that can be described as being true or false. And each sentiment is the sort of thing that could plausibly be in the mind of the speaker who states “The Church is true.”