What are the names of the (semi)secret societies — any and/or all of them — that have influenced, are influencing (and, via common consent, may influence in the future) Buddhism?
The Dhamma(teachings of the Buddha) is the sole authority on Buddhism and the Sangha are the guardians of it. There is no need for secret societies in Buddhism. It is not supposed to be secret or hidden. The Dhamma has the following 6 qualities:
Svakkhato: The Dhamma is not a speculative philosophy, but is the Universal Law found through enlightenment and is preached precisely. Therefore it is Excellent in the beginning (Sila — Moral principles), Excellent in the middle (Samadhi — Concentration) and Excellent in the end (Panna — Wisdom),
Samditthiko: The Dhamma is testable by practice and known by direct experience,
Akaliko: The Dhamma is able to bestow timeless and immediate results here and now, for which there is no need to wait until the future or next existence.
Ehipassiko: The Dhamma welcomes all beings to put it to the test and to experience it for themselves.
Opaneyiko: The Dhamma is capable of being entered upon and therefore it is worthy to be followed as a part of one's life.
Paccattam veditabbo vinnunhi: The Dhamma may be perfectly realized only by the noble disciples who have matured and enlightened enough in supreme wisdom.
It is like the moon. The more it is revealed to the world, the brighter the world becomes.
In DN 16 (below), the Buddha said that he did not hide anything with the closed fist of a teacher who keeps some things back. He did not separate some doctrines as exotic or esoteric from other doctrines.
So this removes the need for secret or semi-secret societies in Buddhism. In fact, all that the Buddha taught can be found online in English, barring those which have not yet been translated to English or not yet digitally transcribed.
So, we should openly reject any groups that claim to teach some secrets in Buddhism that is accessible only to those who join those organizations and swear to keep secrets.
Thus spoke the Venerable Ananda, but the Blessed One answered him, saying: "What more does the community of bhikkhus expect from me, Ananda? I have set forth the Dhamma without making any distinction of esoteric and exotic doctrine; there is nothing, Ananda, with regard to the teachings that the Tathagata holds to the last with the closed fist of a teacher who keeps some things back.
In SN 56.31 (the parable of the Simsapa leaves), the Buddha said that he taught only part of what he directly realized and did not teach a lot of other things. But he said that the things he did not teach were not useful to the path to end all suffering.
Mara and his host, inwardly and outwardly, as for the Dhamma-Vinaya. As for Buddh-ism, it's not secret that it's born from trends of self-identification.
They, those unknowingly or knowingly destruct the Dhamma-Vinaya are usually of pseudoliberal, marxistic views like "The more it is revealed to the world, the brighter the world becomes." Dhamma is something that no-one can reveal for others, yet some might be ready to follow the Noble track. Others, not capable letting go of house (identification) are using all ways to pull it into their outer lands domain.
If this would be true, that this times of easy access would not be the most destructive and harming times since the last Buddha arose. What is not seen or ignored? That this Dhamma is naturally just fir those ready and willing to be tamed and not thought for common folk.
Can you see Mara grasping after the moon?
[Note: this is a gift of Dhamma, not thought for trade, stakes, exchanges or other gains subject toward decay and should be deleted if it's not giften to give in Dhammic conditions]