Translation of the Book of Mormon
One currently popular narrative is that Joseph and Oliver said only that the translation was done by "the gift and power of God," but that's not accurate because it omits the rest of what they said. They both also emphasized that Joseph translated by means of the U&T that came with the plates. They were explicit about that, particularly after the 1834 book Mormonism Unvailed put forth the SITH narrative.
The U&T was essential to the divinity of the translation. SITH not only contradicts what Joseph and Oliver said, but it opens the question of the source of the words that supposedly appeared on the stone. This is why non-LDS Christians say that any book that came from an occult practice (i.e., SITH) could not be divine.
Another popular narrative among some LDS historians seeks to reconcile the accounts by theorizing that Joseph used the term Urim and Thummim to include both the seer stone and the interpreters that came with the plates, but that idea contradicts the plain language Joseph and Oliver used, as well as the common usage at the time.
For decades, those who accepted Joseph Smith as a prophet reaffirmed his claim that he translated the plates by means of the Urim and Thummim, while critics said Joseph actually used the stone-in-the-hat (SITH) method.
In recent years, some faithful LDS scholars have embraced the SITH narrative. Some have concluded that Joseph and Oliver deliberately misled everyone about the translation. To make informed decisions, readers should review the accounts for themselves.
Joseph Smith made several definitive statements about the translation, clarifying that he used the Urim and Thummim (U&T) as instructed by Moroni. At times he and others referred to the U&T as "spectacles" or "interpreters." Those statements that postdate the 1834 book Mormonism Unvailed can be read as responses to the SITH claim in that book.
Responding to ongoing confusion about the translation, Joseph Smith answered the question in the Elders Journal in 1838, unambiguously identifying the instrument he used as the one that came with the plates.
A few people claimed Joseph did not use the plates or the Urim and Thummim, but instead put a seer stone into a hat and read words that appeared on the stone (SITH). This stone, described as about the size and shape of a hen's egg, was dark brown with light brown striations. Joseph had found in a well years earlier.
For detailed analysis of the SITH accounts, compare From Darkness Unto Light, which argues that Joseph used SITH to produce the Book of Mormon, with A Man that Can Translate, which argues that Joseph did not translate with the seer stone but instead used it to demonstrate the process to a few of his supporters.
Aside from the discrepancies in the historical accounts, there is extrinsic evidence that Joseph Smith actually translated the plates. One of these is the lexicon in the text he dictated. Much of it draws from the King James Bible. There are extensive direct quotations, but also many examples of Joseph blending different biblical verses into a single verse in the Book of Mormon.
The Book of Mormon also contains considerable non-biblical language, including words and phrases. Many if not most of these appear in the work of Jonathan Edwards, the influential18th century minister and author who died in 1753. An 8-volume edition of Edwards' works, published in 1808, was on sale in the Palmyra book store that Joseph visited weekly to get the newspaper for his father. Thus, Joseph had access to the non-biblical lexicon we find in the Book of Mormon--just as we would expect if Joseph actually translated the plates, as he said. For more information, see https://www.mobom.org/nonbiblical-intertextuality-database.