Cultural/Translation: Mormon mentions three types of public gathering points, temples, sanctuaries, and synagogues. He immediately follows this with "which were built after the manner of the Jews." This is problematic in that the particulars of the translation do not match with the historical facts. There were no synagogues at the time of Lehi, therefore there would be no Jewish model upon which they might be built. We do not know what kind of building a "sanctuary" was, and indeed, it may not have been a type of building at all, but rather a location. We do know, however, that the first temple erected in the New World by Nephi was built on a Jewish model (2 Nephi 5:16).
In all of these cases, however, it is difficult to understand how a Jewish model might have survived. Temples were certainly following Mesoamerican models, if we have any understanding of any of the archaeological sites where the Book of Mormon might possibly have taken place. We are left, then with a problem of terms. As far as "after the manner of the Jews" is concerned, we really have no idea what such a phrase might have meant to Mormon, writing a thousand years distant from any contact with "the manner of the Jews." If we suppose that his original source had this line, then we still have nearly a 500 year separation from a direct influence. The simplest answer to this particular phrase is to see it as indicative of a style rather than a blueprint. In modern architecture, there are modern buildings that are clearly modern, but which have decorative themes that evoke a more ancient style. Perhaps this is the meaning of "after the manner of the Jews."
It is still possible that we are seeing not a specific translation of the source, but an interpolation of Joseph, an artifact of his translation method. As we have seen in many other cases, Joseph was not familiar with the specifics of the Mesoamerican world, and appears to have interpolated in his own understandings upon the text on the plates. This may be another of those cases, we cannot tell.
What we can tell, however, is that there are three types of sacred places for the Nephites. We understand temples, at least we have access to much information on the ancient temple. We must remember that these locations may share a name with the modern temple, they were quite different in function. In the Book of Mormon we frequently see them as gathering places for public speeches, a function not possible with modern temples, but well suited to the ancient Mesoamerican temple with its courtyard and elevated platforms upon which a speaker could be easily seen.
A synagogue would be a term for a physical building of meeting. In this case, while we do not know of a particular counterpart, it is not surprising to find such a functional location among the Nephites, now that there are churches which have a worship procedure that is separate from the larger culture and society. It is precisely in these conditions of separation from the larger community that such locations are required. Thus while the name might be anachronistic, the function is completely logical.
Sanctuaries are much more difficult to discuss because we have no information on how they were used. In Israel there were sacred locations, usually groves, in the hills. Mesoamerican religion had a similar veneration for sacred spaces in nature, and it is possible that there is a confluence of ideas in the sanctuary that describes such a sacred space in nature. This might be marked with an altar, but might not have any permanent structure at all. At least one sanctuary does have an altar, as witnessed in Alma 15:17. We have two references to building sanctuaries (Alma 21:6, 22:7) though we are not told what needed to be built. Sanctuaries are always places of gathering, however, so even if they are "in nature" they would have required clearing of land sufficient to gather, a process that may be construed as "building."