Christ prescribes the exact words to be used in the ordinance. However, the instruction we use today is slightly different in wording, but identical in meaning: Instead of: "Having authority given me of Jesus Christ" we say instead: "Having been commissioned of Jesus Christ." (D&C 20: 73.)
Why does the authorization come from Jesus Christ, but the ordinance get performed "in the name of the Father, and of the Son, and of the Holy Ghost?" The power to do the ordinance comes from the Son, but the ordinance is in the name of each member of the Godhead. Why?
Though they are one, the names show they occupy different roles and hold different responsibilities. (As to following and being sanctified by different laws and receiving different kingdoms, see D&C 88: 21-26.) We are in the fallen world where the primary means God communicates with man is through the Holy Ghost. (D&C 14: 8.) When, however, a person rises up through the merits of Jesus Christ to receive Him as a minister, they are living in a Terrestrial law and inherit Terrestrial blessings. (D&C 76: 77.) When He has finished His preparations with the person, and can bring them to the Father, the person is brought to a point where the Father can accept and acknowledge them as a son. (See D&C 76: 54-59, 92.) They are then begotten of the Father. (Psalms 2: 7.) Through each of these steps, does baptism matter? Does one receive the companionship of the Holy Ghost without baptism? Do they come to Christ without baptism? Do they inherit what the Father has without baptism? Is baptism critical to the association with each member of the Godhead?
The point at which the person's journey is completed, and they may enter into the rest of the Lord is when the Lord declares by His own voice that the man's offering has been accepted and they are sealed up to eternal life. I've explained this on the blog as to Joseph Smith. I've explained it for Enos and others in Beloved Enos. The Gospel is the same now, as always before. Therefore, no matter how you will receive blessings of the Lord in the afterlife, it will be through the Gospel of Jesus Christ and by the ordinances instituted for claiming blessings. These were established as law to govern man's conduct here even before the world was. (D&C 130: 20-21.)
Note also the person cannot receive the ordinance without also having their name stated. Why do you suppose it is necessary to first call out the name of the person before they receive an ordinance? Why would the Lord's instruction require a person to be "called" first? Though they are submitting to the ordinance voluntarily, why call their name?
Does it matter if the full legal name is used? We do that in the church, of course. But does it matter? If the Lord called Joseph by name at the time of the First Vision (and He did, see JSH 1: 17), what name do you suppose was called? Was it "Joseph Smith, Jr."? Or was it "Joseph"? Or was it that name used by his most intimate friend at the time?
Whenever a name is given by an angel in an appearance to parents, the name is always the first name, or the name their friends would call them. (See, e.g., Luke 1: 13; Luke 1: 31.) Similarly, when the Lord calls a man's name, He uses his first, given name. (See 1 Samuel 3: 4; Exodus 3: 4.) The Lord does not use formal names, but uses intimate names when addressing His servants.
We call the person to be baptized by name. Our practice is to use the full, legal name.