JOEL - Joseph Smith was born December 5 1805. The official story of Joseph Smith reports him saying that in the year 1820 he was at this time in his fifteenth year (JSH 1:7), which means that he was 14 years old at the time of the first vision. So some confusion may come from how people report his age (14 years old, or in his 15th year).
In September of 1823 he had his next vision of the prophet Moroni which happened in his 18th year(or when he was 17 years old).
There were some articles written by various church members which gave conflicting reports on the age of Joseph Smith ranging anywhere from 14 to 17 years old.
William Smith's 1883 account seems to confuse the first vision with the first visit of the angel Moroni, which did occur when Joseph was 17 (WILLIAM SMITH ON MORMONISM, This Book Contains a True Account of the Origin of the Book of Mormon, Printed at Herald Steam Book and Job Office, Lamoni, Iowa. 1883).
William was around seventy years old at the time he published this account; he was about ten at the time of the vision, and as he says, he attended the meetings with the rest, but being quite young and inconsiderate, did not take so much interest in the matter as the older ones did. He also said that Joseph's own acount was more accurate. Its hard to attribute an age inconsistency to Joseph, based on a tract William wrote at seventy years old about things that happened when he was only ten.
The numbers 5 and 6 could very easily be misread and therefore could have been incorrectly printed in a pamphlet or article. For example in an 1832 account the phrase "in the 16th year of my age", was later added above the regular line of script to be inserted into the text when printed. It is possible the printer misread a 5 as a 6.
Other reports say things like "when about fourteen years of age" or "about fourteen or fifteen years of age". If he was "about 15" years of age then he would have been in his 16th year.
The main thing to consider is that Joseph kept no journal at the time of his visions. When he started writing about it 12 years later, he could only give a reasonably accurate guess about his age at the time they happened, which I am sure at the time he was writing, did not seem like a very important detail. If he knew critics were going to make such a big deal about it 150 years later he might have tried to be a little more accurate.
The official account states that Joseph saw God the Father and Jesus in the first vision and the angel Moroni in the next visions. There were several reports of the angel Joesph Smith saw as having been called Nephi, instead of Moroni. This was an error that first showed up in the "Joseph Smith's Manuscript History" published in the Times and Seasons in 1842. At this time, because of Joseph Smith's heavy involvement in politics and being subject to a trial in Missouri, he turned over the editorial responsibilities of the periodical to John Taylor in 1842, and did not get a chance to read and correct the mistake before the article was published. All others who reported on the visit of the angel, calling it Nephi, used the Times and Seasons article for their source of the story and therefore perpetuated the error. It even showed up in an early edition of the Pearl of Great Price, because that part of the story was taken directly from the erroneous Times and Seasons article.
It is not for certain known why John Taylor used the name Nephi instead of Moroni, but consider what else he later said:
"Who was it that administered to Joseph Smith? Moroni and Nephi, men who had lived upon this continent." (Journal of Discourses 19:82).
And he also said:
"the Angel Moroni came to him and revealed to him the Book of Mormon, with the history of which you are generally familiar, and also with the statements that I am now making pertaining to these things. And then came Nephi, one of the ancient prophets, that had lived upon this continent, who had an interest in the welfare of the people that he had lived amongst in those days." (Journal of Discourses 21:161).
So John Taylor confirms that Moroni did appear and revealed the Book of Mormon, but added that Nephi also appeared to Joseph Smith at some unknown time. There is no other supporting evidence however that Nephi actually did appear to Joseph Smith. Although an entry in the History of the Church states:
"The angel(Moroni) appeared to me three times the same night and unfolded the same things. After having received many visits from the angels of God unfolding the majesty and glory of the events that should transpire in the last days, on the morning of the 22nd of September, A.D. 1827, the angel(Moroni) of the Lord delivered the records into my hands." (HC 4:537)
Here Joseph states that he received "many visits from the angels", one of which could have been Nephi.
There are several other earlier sources that verify that it was Moroni who visited Joseph Smith in connection with the bringing forth of the Book of Mormon(See for example D&C 27:5, written in 1830, and D&C 128:20 written in 1842)
Joseph Smith himself also identified the angel in an 1838 report:
"Question 4th. How, and where did you obtain the Book of Mormon?
Answer. Moroni, the person who deposited the plates, from whence the Book of Mormon was translated, in a hill in Manchester, Ontario County, New York, being dead, and raised again therefrom, appeared unto me and told me where they were and gave me directions how to obtain them. I obtained them and the Urim and Thummim with them, by the means of which I translated the plates and thus came the Book of Mormon. (Elders Journal 1, pp. 42-3 (July 1838)