The Church associated with the "Second Invitation" is named the Church of Jesus Christ of Latter Day Saints, which is the same name which a revelation to Joseph Smith stated that his church would be known by in the latter days (D&C 115:4 original). This Church is based in Brazil and is not the corporate church based in Salt Lake City, Utah, which has changed the spelling to use a hyphen in "Latter-day".
The Brazilian based church is the continuation of the original church headed by Joseph Smith. In effect, it is the restoration of the restored church. This return back to the fundamentals of the Church was necessary because the modern corporate church, as well as the branches of the original church in Missouri, have all strayed so far from the original that simply upgrading their leadership could not have sufficed to bring them back on track.
Thus, the angels Raphael and Moroni have appeared to a new prophet in Brazil, provided him with the same golden plates, which Joseph Smith translated into the Book of Mormon. He has translated some of the sealed plates into what has been published as The Sealed Book of Mormon, which is now downloadable as a PDF file from this website.