|G. Washington, "President of America"|
A lot of the confusion comes from statements such as this one from Wilford Woodruff's journal, dated April 1844:
"Conference met at 10 oclok April 8th. President J Smith arose and said it is impossible to continue the subject that I spoke upon yesterday in consequence of the weekness of my lungs. Yet I have a proclamation to make to the Elders. You know the Lord has led the Church untill the present time. I have now a great proclamation for the Elders to teach the Church here after which is in relation to Zion. The whole of North and South America is Zion. The mountain of the Lords House is in the centre of North & South America."
Mesoamerican activists actually cite this as evidence that the Book of Mormon took place in Central America.
Others read the rest of the journal entry and get a better idea of what was intended.
"When the House is done, Baptism font erected and finished & the worthy are washed, anointed, endowed & ordained kings & priests, which must be done in this life, when the place is prepared you must go through all the ordinances of the house of the Lord so that you who have any dead friends must go through all the ordinances for them the same as for yourselves; then the Elders are to go through all America & build up Churches until all Zion is built up, but not to commence to do this untill the Temple is built up here and the Elders endowed. Then go forth & accomplish the work & build up stakes in all North and South America. Their will be some place ordained for the redeeming of the dead. I think this place will be the one, so their will be gathering fast enough here."
In modern times, we interpret this to mean the continents of North America and South America, but that's now how it was meant in 1844.
About a year later, in June 1845, Woodruff was back in England. He visited the exhibition of Madame Tussaud and Sons. One of the exhibits that most impressed him was this:
"George Washington dressed as the President of America taken from A bust executed from Life. This personage bespoke as much dignity as any member of the Group."
Woodruff used the term "America" interchangeably with the United States, or the United States of America. He referred to it as America three times as often as he did the United States. Like his contemporaries, he was concerned about the division between the northern states and the southern states. For example, he made this comment in Volume 2 of his journal:
"After General Harison was elected President of the United States A body of citizens suspended a line across the road in which the President was to walk. This line contained or supported 27 flags one for each of the states. As General Harrison was passing under thes colors the line parted in the centre. One half fell into the street towards the north & the other half towards the south as much as to say the states would be divided."
When we think about Joseph Smith's statement from the April 1844 conference, he was referring to the Nauvoo temple, which was built "in the centre of North and South America" if we're referring to the United States of America. It's nearly as central as possible in a literal sense, given the unknown extent of the western territories.
When Joseph said the Elders were to go through "all America" "& build up stakes in all North and South America," he was referring to the United States, as we can see not only from the ordinary use of the term "America" at the time, but also from the reality of what actually happened. The Elders were already in Europe. They didn't go to South America until much later. But it was important for the members to know they would build the Church in both North and South America. They were not going to focus just on the northern states, where most of them had come from.
The question remains, how could there be descendants of Lehi throughout the western hemisphere if the events of the Book of Mormon took place in North America (using modern terminology).
The quick answer: the Nephite civilization was destroyed in western New York, after years of battles all the way from Zarahemla in Iowa, across the midwest (Bountiful) eastward to New York. Later, after the Book of Mormon record was concluded and Moroni buried the plates, people from the Mayan civilization migrated northward and occupied what is now the Southeastern U.S. and the Mississippi River valleys. After a few hundred years, they left and returned to their homeland in Central America. This explains how Lehi's blood, however diluted, made its way throughout what today is known as Latin America.