'...in his dreams...'?
(Note: This essay is "lifted" with very little editing from the main part of the website, so isn't specifically about Tolkien's faith. But it has some comparisons/contrasts between what I would call Frodo's and Sam's spiritualities that might be useful.)
A bit different from each other, aren't they? Small differences, perhaps, but not insignificant (or Tolkien wouldn't have bothered with them). I see two aspects of the characters coming into play here: Frodo's identity as a visionary, and both Frodo's and Sam's personalities as reflected by how they express things in their respective sections of the Red Book.
Frodo has several dreams that we're told about in detail, and more that are only briefly referred to. At other times, he has what we might call mystical experiences or even visions. This description from "Fog on the Barrow Downs" is especially interesting because Frodo himself isn't sure if it's a dream or something else: "...either in his dreams or out of them, he could not tell which..." The parallel that first comes to my mind is Paul's description in his second letter to the Corinthians (12:2-3) of his vision of heaven, where Paul, speaking in third person to lessen his "boasting," says he "... was caught up into Paradise - whether in the body or out of the body I do not know, God knows..." That uncertainty would almost be expected in a mystical experience, where the person's awareness extends beyond what the physical senses, or even the normal dream state, would provide.
Frodo's aim in his writing was to record what happened to him and to his companions so that future generations wouldn't take their peace and happiness for granted. Since he doesn't make the Red Book available to anyone until immediately before he leaves Middle-earth, he doesn't have to worry about protecting his pride, or his humility, in what he's written. He's free to tell the truth. There are a number of passages (and this is one of them) in which he seems to search for words that will get as close to the truth as possible, even though the truth is beyond words. One way to recognize these passages is to look for paradoxes - especially paradoxes we might not notice right away, because something inside us understands them well enough that we're not immediately aware of the seeming contradiction. One of these comes in Mordor, when Frodo tells Sam that he is "naked in the dark" but that there is "no veil between me and the wheel of fire." Those phrases, when taken together, don't make literal sense; how can someone be in the dark and yet have his sight filled by a fire? But I've never heard of anyone trying to make literal sense of the paradox Frodo uses there; we know he's attempting to describe a state of being that's beyond literal explanation.
Besides Frodo's statement that he doesn't know if he was dreaming or in some other state of awareness, there's another aspect of this passage from "Fog on the Barrow Downs" that tells us the vision was beyond what he could put into words. Some of the hallucinogen-using readers during the 1960's may have been able to take this description literally, but the rest of us would have a hard time of it: "...a song that seemed to come like a pale light..."
Is it a song that's seen as a light, or light that's heard as a song? Or something else entirely? The incongruent reference to two different senses implies, I believe, that what Frodo is experiencing isn't something that can be captured by any one of our physical senses. Again, he's doing his best to find a way to describe in words an awareness that goes beyond words, and what he comes up with is a combination of hearing and sight, as if during this experience they became a single sense - a much richer sense than either of them taken alone.
And this song made of light, or light made of song, or something else entirely, isn't there just to be noticed. It affects the reality around it. That adds another layer to the mystery, because we don't really know what it's affecting or what changes it's bringing about - except by words that both Frodo and we know are limited. The song seems to come like a pale light behind a grey rain-curtain, which I take to mean that even the rain-curtain is a metaphor for something else. And if we look at the sentence structure and take out separating phrases, we notice that the song "seemed to... turn the veil all to glass and silver, until at last it was rolled back." (Emphasis added.) Even the description of the veil as glass and silver hints that neither comparison is exact.
So, one thing that Frodo definitely says is real is the song. The other elements he mentions in connection with it are comparisons to things he hopes his readers will understand. "Frodo heard a sweet singing running in his mind: a song that seemed to..." Nowadays that should set bells ringing in the mind of anyone who's read The Silmarillion. When LotR was published, The Silmarillion wasn't yet available and the general reader of LotR didn't know that, in Tolkien's mythology, the world was created through song. But, of course, Tolkien knew, and everything he wrote in LotR he made consistent with the wider world and longer history that he knew existed. So if one thing is going to be real in Tolkien's cosmos it will be a song - the Song, although Frodo might not have understood all the implications. But, of course, Tolkien did.* In fact, in Splintered Light, Verlyn Flieger says "Light and music are conjoined elements made manifest in the visible world as the Music of the Ainur." So the joining of a song with light makes perfect sense.
The other part of the vision that Frodo indicates is what he says it is - rather than a metaphor for something indescribable - is the "far green country." On the other hand, he doesn't give us any idea of just what the far green country is. But, then, that's how Frodo normally recorded his dreams: he described them but didn't try to interpret them, at least not in the material Tolkien passes on to us from the Red Book. The only exception I can think of is his blurting out, "I saw you!" to Gandalf at the Council of Elrond. But then he becomes unsure of when he'd had the dream, and Gandalf tells him the timing was wrong... Maybe that's why he didn't try again.
But, certainly, the far green country is a vision of the Undying Lands... isn't it? Maybe, but it's not Frodo who interprets it that way for us, but Sam.
There's one important element we find only in Sam's version of the dream: the white shores. Frodo doesn't mention them at all; in fact, his description doesn't even hint that the Sea itself comes into the dream/vision (although we know it's come into many of his earlier dreams). There are a couple of intriguing possibilities for the reason Sam made this addition. My original supposition was that Sam added it of his own volition, to make his picture of hope for Frodo even stronger and more concrete. As a student of Bilbo, Sam would almost certainly have known about the sand made of diamond dust the legends say Eärendil found on the shores of Valinor. Since Sam had already decided to use what he knew of Frodo's dream in composing the end of his story, adding the white shores would have been a small step. Another possibility, though, is that Frodo told Sam more about that dream than he wanted to tell all the future readers of the Red Book. We don't know that Frodo didn't see white shores in his dream/vision, or that he wasn't aware of the Sea during it. If Frodo had told those details to Sam, Frodo's sailing would certainly bring them back to Sam's mind - to the extent that the ending he wrote to the story becomes a likely interpretation of the dream, rather than just Sam's fond hopes for Frodo's future.
Sam teases apart the senses in his version; we don't see any music or hear any light, as we do in Frodo's. Sam even adds the sense of smell, so the adjective "sweet" can be applied to a scent rather than something heard. Although most people wouldn't be confused by the phrase "sweet singing," I think this change goes along with Sam's personality. As a "sensate" (according to the Myers-Briggs Personality Inventory), Sam is more conscious of the physical world around him and of exactly how he comes to know it through his individual senses than would be an "intuitive" such as Frodo, who would be more conscious of the overall effect of what he experiences than he would be of the individual details, and even less aware of which detail is coming to him through which specific sense. Each author is going to write according to the way he experiences things. In addition, if Sam is trying to emphasize that the ending he's writing to the story is true, he would be likely to make it as concrete as possible. Frodo's "glass and silver" become Sam's "silver glass," which isn't a bad description of sunlight shining through a curtain of rain just before the clouds break (or the rain curtain is "rolled back"). And Sam doesn't connect the rain-curtain's changing to silver glass, or its rolling back, to the song itself as Frodo does; the singing Sam tells us about is just singing - not a force that affects the reality around it. The way Sam words things, it's much easier to take the whole description as a literal report of what's happening in the physical world, which is where his hopes for Frodo lie.
The experience Frodo describes, on the other hand, could be a foreknowledge of something that will happen to him within the bounds of Arda, but it may be something else. All of it, or some of its elements, may belong to Frodo's inner life or to the reality beyond physical creation to which his inner life - as a mortal - is bound.
*To see how the Song of creation would be related to Frodo, thousands of years after the creation of Arda, it's necessary to see creation as a continuing process. The Song doesn't just form the physical world initially, it also plays out its history. As brought up in "Magic in Middle-earth," free-will decisions made by mortals can actually alter the course of the Song.