Close. The main body of Mormons was evicted from Illinois in 1846. Those who didn’t follow Brigham Young were left alone. Among those who stayed in the east there arose several who claimed to be Joseph Smith’s true successor (as opposed to Brigham Young.) Granville Hedrick, William Smith, James Strang, Jason Briggs, Zenos Gurley and W. W. Blair each led a faction of the remaining population of Mormons. In 1860, Gurley, Briggs and Blair convinced Joseph Smith III, to accept the presidency of the Reorganized Church of Jesus Christ of Latter Day Saints. The RLDS turned out to be the largest of these groups that opted not to follow the 12 Apostles of the LDS Church with Brigham Young at their head.
Alma
Not quite so. Sidney Rigdon made the first claim as Smith’s successor, based on being the only surviving member of the First Presidency. James Strang claimed visions that indicated he was the successor. Brigham Young claimed the Quorum of the Twelve, as a body, held the keys to the priesthood, and not any one person.
At an August 1844 General Conference, Rigdon and Young presented their positions, and a vote by the congregation sided with Young. Young’s position being that the succession to Smith’s leadership of the church passed onto the Quorum of the Twelve. The dissenters (nay voters) were excommunicated by Young. Young announced via the Times and Seasons that Smith’s followers no longer had a prophet, but a Quorum.
Sidney Rigdon did not abandon his claim, and continued to build his own followers. He was excommunicated by the Brigham Young faction, and he fled to Pittsburgh, in fear of Young and his followers. Rigdon’s followers, from Nauvoo, followed him there. In 1845 he claimed, officially, to be the successor, and named himself President of the church (Smith’s church).
Meanwhile, James Strang built up followers by claiming to be a prophet (in contrast to Young’s announcement that there was no longer a prophet, but a quorum). He attracted those who were attracted to the claim of continuing prophecy. His followers, from Nauvoo, followed him to Wisconsin, including Joseph Smith’s last surviving brother, William.
Young reorganized a new First Presidency in December of 1847, in Utah, with himself as the new President. The Rigdon and Strang churches failed, though the both have a small group of loyal followers today.
Many who remained in the Midwest, believed the rightful successor to be Joseph Smith’s son, Joseph Smith III., who Smith Jr. had annointed at least three separate times as his named successor. At the time of Smith’s murder, the boy was too young to lead, thus, the succession crisis. After Smith III had grown up, the various groups of Mormons, including Young, approached Smith III and his brother David Hyrum, asking them to join the leadership of their respective churches. Smith III said he would not, unless inspired to do so, and more particularly, would not join the Brighamite group because of their practice of plural marriage.
Smith III, in 1860, agreed to lead a group of Midwest Mormons, who had coalesced under the leadership of Jason W. Briggs, Zenas H. Gurley and William Marks, as the Church of Jesus Christ of Latter Day Saints. His mother Emma and his three brothers became affiliated to this group. This church later added “Reorganized” to the church name (RLDS), and this name was changed much later to Community of Christ.
. I think most Christian groups today, even though they inherit and essentially agree with the established Catholic position on the Trinity and the Hypostatic Union, still often struggle to explain them and often describe them in terms that fit more into the more nuanced heresies, like Monothelitism and Monophysitism.