Short answers, pulled from the story.
Neo-Theosophy is a term, originally used as an insult by followers of Helena Blavatsky, for the Theosophical ideas that Annie Besant and Charles Webster Leadbeater developed after Blavatsky died in 1891. Many Theosophists around the world nonetheless accept these ideas as genuinely Theosophical.
F.T. Brooks coined the term Neo-Theosophy in 1914 in his book Neo Theosophy Exposed, the second part of an earlier work called The Theosophical Society and its Esoteric Bogeydom.
William Quan Judge disputed with Henry Steel Olcott and Annie Besant over accusations that he had forged letters from the Mahatmas, and in 1895 he broke away with most of the Society's American Section. The Olcott and Besant faction, based in India as the Theosophical Society - Adyar, became the group its critics label Neo-Theosophy.
Besant and Leadbeater introduced clairvoyant investigation into past lives and the astral plane, promoted the young Indian boy Krishnamurti as the vehicle of a coming World Teacher, and brought Catholic ritual into the movement through the Liberal Catholic Church.
Robert S. Ellwood's 1973 book Religious and Spiritual Groups in Modern America grouped Neo-Gnostic and Neo-Rosicrucian groups, Rudolf Steiner's Anthroposophy, Alice Bailey's groups, Guy Ballard's "I AM" Activity, and Max Heindel's Rosicrucianism under this label. Daryl S. Paulson later linked the term specifically to Alice Bailey.
Benjamin Creme, Douglas Baker, and Victor Skumin are cited as examples of Neo-Theosophists today. In 1990, Skumin proposed a classification called Homo spiritalis, Latin for "spiritual man," describing it as the sixth root race.