First I consulted Google about the different accounts and I found: https://www.lds.org/topics/first-vision-accounts?lang=eng. The main differences between the two different accounts is that in the first, which Joseph wrote in a private place never intending it to be published, Joseph only mentions seeing the Lord. Notice the use of the singular. In the second, Joseph records that he saw two personages.
The article gives various possibilities for the differences inbetween the two accounts. Joseph could have merely been emphasizing the part of his vision when Jesus, the Lord, talked to him. Or he could have also referred to both individuals as the Lord.
There are other, more consistent ways of seeing the evidence. A basic harmony in the narrative across time must be acknowledged at the outset: three of the four accounts clearly state that two personages appeared to Joseph Smith in the First Vision. The outlier is Joseph Smith’s 1832 account, which can be read to refer to one or two personages. If read to refer to one heavenly being, it would likely be to the personage who forgave his sins. According to later accounts, the first divine personage told Joseph Smith to “hear” the second, Jesus Christ, who then delivered the main message, which included the message of forgiveness.10 Joseph Smith’s 1832 account, then, may have concentrated on Jesus Christ, the bearer of forgiveness.
Another way of reading the 1832 account is that Joseph Smith referred to two beings, both of whom he called “Lord.” The embellishment argument hinges on the assumption that the 1832 account describes the appearance of only one divine being. But the 1832 account does not say that only one being appeared. Note that the two references to “Lord” are separated in time: first “the Lord” opens the heavens; then Joseph Smith sees “the Lord.” This reading of the account is consistent with Joseph’s 1835 account, which has one personage appearing first, followed by another soon afterwards. The 1832 account, then, can reasonably be read to mean that Joseph Smith saw one being who then revealed another and that he referred to both of them as “the Lord”: “the Lord opened the heavens upon me and I saw the Lord.”11In order to read the different accounts and explain the differences between them, you must first choose one of two axioms. Either Joseph Smith told the truth in all the accounts, in which case he was a prophet of God, or Joseph Smith lied in one or more accounts, and then it is reasonable to assume that he wasn't a prophet of God.
Using just the various accounts of the first vision, it is impossible to tell which axiom is true. Both can be explained using simple logic and it comes down to an interpretation of what Joseph Smith meant when he wrote it. Anyone who know me, and knows how much I struggle with literature classes, will understand that this is a question I try to avoid as much as possible. I don't know what the author meant when he/she wrote something. It is impossible to say. All you have is the words on the paper, and they can mean whatever you would like them to mean. You can not say for certain what the author meant when he/she wrote something. The only way to know for certain what the author meant is to ask him/her and be reasonably assured that the author is telling the truth when he/she responds. Unfortunately Joseph Smith is dead, so we can't ask him what he meant. (Imagine how great that would be, if when reading a particular difficult text, like Isaiah or Shakespeare, instead of just grasping at straws trying to understand the meaning, you could just call the author up on the phone and ask them. I think that would solve every student's problems in literature classes instantly.)
So just looking at the accounts of the first vision, there is no way to know what is true. Either Joseph Smith lied or he told the truth. Either he was a prophet or he wasn't. Choose your axiom, and you can defend it either way. This corresponds with an agnostic view towards the world. You can't know, so stop trying to know because you will just keep getting into circular arguments.
Fortunately, we have more resources available then just the original sources of the first vision and the various commentaries on it. Joseph Smith translated (or made up, depending on the axiom you choose) a work of ancient scripture, the Book of Mormon. This is the only book on the face of the earth with a promise of its kind in it. The book promises that if you will read the book, ponder its message, and ask God, then you will come to know the truth of it.
We invite all men everywhere to read the Book of Mormon, to ponder in their hearts the message it contains, and then to ask God, the Eternal Father, in the name of Christ if the book is true. Those who pursue this course and ask in faith will gain a testimony of its truth and divinity by the power of the Holy Ghost.Suppose that Joseph Smith told the truth. (As anyone who has taken mathematical analysis has practiced, it is possible to consider that something is true, without actually accepting it as truth, and then seeing where it leads.) Then there is a reasonable explanation between the differences of the accounts. Then Joseph Smith was a prophet of God, and the promise made in the Book of Mormon is a promise from God. Then you can pray to know the truth, and God will answer you.
This is as far as my proof can go. Unlike mathematical proofs, I can not demonstrate it for you, and stop my argument with a QED. Only God can prove it to you. You have to read the book yourself, and then ponder in your heart it's message. Then you must ask God, in faith, and you will come to know for yourself that it is true.
If Joseph told the truth, then he wasn't the only witness to the events surrounding the first vision, and although we can't ask Joseph what he meant by the differences in the accounts, we can ask God if they are true. This is a much better practice of obtaining truth then by trying to understand what the author meant when he wrote it.
I have asked God if it is true, and He has answered me. But before I get to my witness, I want to share the testimony of Elder Holland, a special witness of Christ.
I know that the Book of Mormon is true. I know because I have prayed and asked God if it is true. And every time I get the same results. In a way, more powerful than words, I know that it is true. The Holy Ghost testifies to me of its truthfulness. In fact, as I am writing these words, again I am receiving a witness from God that it is true.
I don't know how to reconcile the differences between the different accounts of the first vision, but I know that the Book of Mormon is true, thus I know that Joseph Smith was a prophet, and he didn't lie. He did see God the Father and his Son Jesus Christ. And the church that he established is the Kingdom of God on Earth once more.
If you want to know of it the truthfulness of the Book of Mormon, if you want to know if Joseph Smith told the truth, or if you want to know if the Church of Jesus Christ of Latter-day Saints is the Kingdom of God, then all you have to do is ask. Start by supposing that it is true (remember that you can suppose something is true, before you have established it as fact -- mathematicians do it all the time), then read the book, and in faith, ask God. You will get a witness from the Holy Ghost.
(Note that it is super important to start by supposing the Book of Mormon is true, because then the promise in the Book of Mormon is from God, and it is a real promise. If you start by supposing the Book of Mormon is false, then the promise in the book was made up by a liar, and you can't do anything about it, and it won't lead you to any more conclusions.)