TERRY - What exactly is our "as a church" believe to be the definition of "Special Witness"? I have heard members of my own ward and some (not all)throughout my life that it meant "literal".

JOEL - The term "special witness" is reserved for those who are called to be an apostle of the Lord Jesus Christ in all the world. They know of the reality of Christ and his redemption with a certainty born of the Spirit. As a special witness it is their solemn responsibility to proclaim to the world the existance of and mission of the Lord Jesus Christ.
In one general conference, Elder Boyd K. Packer explained this responsibility:

"We do not talk of those sacred interviews that qualify the servants of the Lord to bear a special witness of Him, for we have been commanded not to do so. But we are free, indeed, we are obliged, to bear that special witness." (In Conference Report, April 1980, p. 86.)

Harold B. Lee also described what is means to be a special witness:

"Jesus lives! I bear my witness to you this morning. There are some witnesses I cannot give now, perhaps sometime later.
Many things are too sacred to share at this time. I have received a witness that I cannot or dare not deny. When I see Jesus, I cannot mistake His identity. I know that He lives!
I came away from some of these experiences never to feel the same again about the mission of our Lord and Savior. I had impressed upon me, as I have never had it impressed before, what it means to be a special witness. I say, with all the conviction of my soul, I know that Jesus lives. I know that He was the very Son of God. And I know that in this church and in the gospel of Jesus Christ is to be found the way to salvation." (Teachings of Harold B. Lee, Harold B. Lee p. 636)

Some latter-day prophets have related experiences of actual visitations of the Lord. President George Q. Cannon related:

“I know that God lives. I know that Jesus lives; for I have seen Him. I know that this is the Church of God, and that it is founded on Jesus Christ, our Redeemer. I testify to you of these things as one who knows—as one of the Apostles of the Lord Jesus Christ that can bear witness to you today in the presence of the Lord that He lives and that He will live, and will come to reign on the earth, to sway an undisputed sceptre.” (Delivered in the October 1896 General Conference and reported in The Deseret Weekly, October 31, 1896, vol. 53, p. 610.)

And one about Lorenzo Snow:

"President Snow put on his holy temple robes, repaired again to the same sacred altar, offered up the signs of the Priesthood and poured out his heart to the Lord. He reminded the Lord how he plead for President Woodruff’s life to be spared, that President Woodruff’s days would be lengthened beyond his own; that he might never be called upon to bear the heavy burdens and responsibilities of the Church. “Nevertheless,” he said, “Thy will be done. I have not sought this responsibility but if it be Thy will, I now present myself before Thee for Thy guidance and instruction. I ask that Thou show me what Thou wouldst have me do.”
After finishing his prayer he expected a reply, some special manifestation from the Lord. So he waited,—and waited—and waited. There was no reply, no voice, no visitation, no manifestation. He left the altar and the room in great disappointment. Passing through the Celestial room and out into the large corridor a glorious manifestation was given President Snow which I relate in the words of his grand-daughter, Allie Young Pond, now the wife of Elder Noah S. Pond, recently president of the Northern States Mission:

“One evening while I was visiting grandpa Snow in his room in the Salt Lake Temple, I remained until the door keepers had gone and the night-watchmen had not yet come in, so grand-pa said he would take me to the main front entrance and let me out that way. He got his bunch of keys from his dresser. After we left his room and while we were still in the large corridor leading into the celestial room, I was walking several steps ahead of grand-pa when he stopped me and said: ‘Wait a moment, Allie, I want to tell you something. It was right here that the Lord Jesus Christ appeared to me at the time of the death of President Woodruff. He instructed me to go right ahead and reorganize the First Presidency of the Church at once and not wait as had been done after the death of the previous presidents, and that I was to succeed President Woodruff.’
“Then grand-pa came a step nearer and held out his left hand and said: ‘He stood right here, about three feet above the floor. It looked as though He stood on a plate of solid gold.’
“Grand-pa told me what a glorious personage the Savior is and described His hands, feet, countenance and beautiful white robes, all of which were of such a glory of whiteness and brightness that he could hardly gaze upon Him.
“Then he came another step nearer and put his right hand on my head and said: ‘Now, grand-daughter, I want you to remember that this is the testimony of your grand-father, that he told you with his own lips that he actually saw the Savior, here in the Temple, and talked with Him face to face.’” (Improvement Era, September 1933, under the title “An Experience of My Father’s” By LeRoi C. Snow:)

So each Apostle has some convincing moment of communication, visitation, vision, or revelation from Christ in a way that they cannot deny nor doubt His existance nor their calling to be His witness. Because of this one could call them "literal witnesses".