Mormonism claims the only "authority" to baptize...something left out of your post.
One must have the proper authority to baptize. Mormons believe that only members of The Church of Jesus Christ of Latter-day Saints who hold the Aaronic Priesthood hold the keys of baptism. The keys were handed down from John the Baptist (who appeared as a resurrected being) to Joseph Smith and Oliver Cowdery when the Church was restored.
Not just anyone can perform baptisms. The power to baptize comes from the Lord. In 3 Nephi 11:19-21, (Book of Mormon) the Lord gives Nephi the power to baptize in His name. The authority must be valid, coming from the Lord."