Foundation essentially means authenticating that a piece of evidence actually is what it purports to be. Relevance means that a proffered piece of evidence actually does tend to make a given claim more likely than it would be without that evidence.
The reason these concepts are invariably left out are that they do not seem to be well understood by most defenders of the faith, and when they are, issues of foundation and relevance are not helpful to the truth claims of the LDS Church.
For example, this Sic et Non post discussed in the OP refers to the supposed witnesses of the Book of Mormon:
Daniel Peterson wrote: The testimonies of the Witnesses to the Book of Mormon are evidence.
We could do a whole thing on the Three Witnesses, but for simplicity I'll just talk about the Eight Witnesses. What did they actually witness?
They saw an object they were incapable of identifying and heard Joseph Smith make a representation of what that object supposedly was. That's it. They were not "witnesses to the Book of Mormon" at all, because they had no way to verify that the plates Joseph Smith showed them were an ancient Nephite record. They had no way to determine that the Book of Mormon was a translation of the inscriptions on those plates, because they could not decipher them. And to believing Latter-day Saints who think it's ridiculous that plates could be fabricated and claimed to be of ancient origin, let me direct your attention to two things: the Kinderhook Plates and the Voree Plates.
The testimony of the Eight Witnesses is not evidence that the Book of Mormon is true or that Joseph Smith's story about the golden plates is true. The Eight Witnesses had no independent knowledge of either of those things. But if your goal is simply to prove that Joseph Smith showed some of his close friends and relatives an object and told them a story about it, then hey, you've got evidence.
There is also this:
There is certainly evidence for the claims of Mormonism. Joseph Smith's own account is prima facie evidence, for example. (We almost always accept first person accounts of an experience if we have no particular reason to reject them.)
Joseph Smith's own account is not prima facie evidence. Prima facie evidence is when, unless there is evidence to refute it, the evidence presented is sufficient to establish a claim. Joseph Smith's own account is direct evidence, because he claims to have personal knowledge of the events at issue, but it is not prima facie evidence.
But Peterson is also begging the question of "Joseph Smith's own account." Own account of what? Of the First Vision? Which of his several and varied accounts of the First Vision is the evidence? It's not the version in the Pearl of Great Price, because that isn't Joseph Smith's own account. He didn't write it---someone else did and put it in the first person.
Mopologist sermons on evidence also omit any discussion of heuristics. It isn't true that we almost always accept first person accounts of an experience if we have no particular reason to reject them. What is true is that whether conscious of it or not, most people generally agree that extraordinary claims require extraordinary evidence. Very few of the readers of this board know who I really am or really know anything about me. Without even needing to know a lot about my life and whom I associate with, which of these claims are you more likely to accept at face value?
(A) I watched a DVD with my kids on Saturday night and we popped popcorn.
(B) I watched a DVD with Scarlett Johansson on Saturday night while she was sitting on my lap.
Now, claims like theophanies can't be falsified in the same way that most claims of fact can. But if I have nothing but a person's word for it, and that person changes his story over time in significant ways (when it happened, why it happened, what happened), then I'm pretty far along in being able to evaluate whether I can rely on his word. And when alleged supernatural experiences involve claims of fact about the tangible world, then those claims become falsifiable.
Supposed Book of Mormon evidences are not evidences of the Book of Mormon narrative. Evidence about the Olmecs, the Aztecs, the Mayans, the Moundbuilders, etc. are not relevant to the Book of Mormon. Contrary to the implication of Peterson's blog post, a reasonable person can say quite confidently that there is no evidence for the historicity of the Book of Mormon. That includes the NHM thing (either of them--the Ensign has talked about two different sites). The NHM idea suffers from both lack of foundation and lack of relevance.
Whether consciously or not, most people also accept Occam's Razor as a valid way to evaluate various claims. Or, put another way, if the Book of Mormon were a 19th-century hoax, it would look exactly the same as it already does. There is not a single proffered piece of evidence in favor of the Book of Mormon's alleged historicity that does not have a more cogent and likely explanation. In Mormon apologetics, however, we have Joseph's Razor: whatever explanation tends to leave the door open that the Church's truth claims could be accurate, no matter how contrived, implausible, or baseless, that explanation is probably true. Again, this is why heuristics are notably absent from Mopologist sermons about evidence. Mopologists are not trying to arrive at a conclusion on the basis of evidence. They are trying to explain away evidence that conflicts with their foregone conclusions.
(I've been talking about the Book of Mormon, but the foundation and relevance problems are applicable to the Pearl of Great Price as well.)
Then there's Peterson's theory that God is like the Riddler from Batman:
The claims of Mormonism are, I think, right about where they're supposed to be: Not so obviously true as to coerce acceptance, and not so obviously false as to make acceptance illegitimate.
Okay, so what you're telling me is that God loves us so much that He'll send His Only Begotten Son to be tortured to death for us, but he'll still play Three-Card Monte with our salvation. He has set up this wonderful plan for us to return to His presence, and restored His church in these latter days so we can know about His plan, but also made sure to set things up so that it's a coin toss whether to believe in the One, True Church or not. (Yes, I'm aware of the mixed metaphors. It's late and I'm tired.)
But even assuming for argument's sake that Peterson is right, and the evidence is a draw, then the rational solution is not to maintain belief in the Church and ignore contrary evidence by "putting it on the shelf." The rational solution is to withhold belief because the burden of proof has not been met. Burden of proof is, of course, another concept that Mopologist "evidence" sermons omit. If the LDS Church's claims of fact regarding the Book of Mormon, the Book of Abraham, etc. have not been proven, it is not the responsibility of the rest of the world to disprove those claims in order to justify non-belief. Nor is providing a counter-explanation the burden of unbelievers. Absence of evidence is a perfectly reasonable and intellectually honest basis for rejecting a claim about anything, Mormon-related or otherwise.
Oh, but wait! To paraphrase something Gadianton said one time (and I can't find now), the Mopologist strategy is to assert that the evidence for and evidence against the Church is at a stalemate, and then the Holy Ghost can be the Trump card that resolves everything in the Church's favor. Moroni's Promise is a whole subject unto itself, but the short answer is this: there's no way to assess the validity of Moroni's Promise. You don't have a way to independently verify that your subjective emotional experience means what the LDS Church tells you it's supposed to mean. I read the Book of Mormon. I pray about it. I feel good. The missionaries tell me this means that the Holy Ghost is telling me that the Book of Mormon is true. In the LDS context, "true" doesn't just mean good principles or that there's a God or something else abstract and/or metaphysical. "True" in this instance means that the Book of Mormon is real history. So how do I know that I'm not being subjected to wishful thinking, to operant conditioning, that I'm delusional, or that I'm misinterpreting an otherwise legitimate spiritual experience? Well, the way to do that is by seeing how what the missionaries told me was an "answer" from the Holy Ghost comports with objective reality. And now we're right back to square one of the evidence problem that Moroni's Promise was supposed to solve.
Evidence never has been and never will be a building block that a Mopologist uses to construct something. Evidence is merely the pet name Mopologists give to the unicorn they are sending you off to hunt.