Which (semi)secret societies influence Buddhism? 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.