Srimate Raamanujaye NAmaha,
Swamy, I quote below few paras from the explanation of Raamanuja nootrandhi (for verse 46 in Sadagoppan .org)
“The six darsanams that NammAzhwAr refers to here have been counted in number of ways to add up to a total of Six: (1) The six religions listed by AchArya RaamAnujA are Bhouddham, ChArvAkam, NaiyAyka-VaisEshikams of GouthamA, Jainam, Saankhyam and Yogam. These six are also known as Saakhya, ulUkhya, AkshapAtha,KshapaNa, Kapila and Pathanjali darsanams based on the people, who developed them or wrote commentaries.
(2) In another counting, these six are recognized as Saankhyam, Yogam, Bhouddham, VaisEshikam, Jainam and Paasupatham.
(3) In yet another counting, these six are: ChArvAkam, Jainam and four types of Bhouddham (VaibhAshikam and SautrAntikam of HeenayAnam and yOgAchAram and nihilism of MahAyAnam sects).
The various darsanams listed above fall into the categories of Naasthika and Aasthika darsanams. The Naasthika darsanams like ChArvAkam (materialistic 141 sadagopan.org atheism), Buddhism and Jainism are systems of thought that do not accept the authority of the VedAs and deny the tattvams such as Soul (Jevan) and God (Isvaran), which Aasthika mathams accept as fundamental to their beliefs. Naasthika mathams are called Veda Baahya mathams and the Aasthika Mathams are saluted as Vaidhika Mathams.”
So there are few variations of what these 6 are.. as you have also indicated a set of 6 above.
Also within each religion, different versions of dwaita, adavaita or vishishadaviata are followed. For example In Shaivism there are certain sects with advaita focus and some very similar to visishta adawaitam. In Vaishnavism also same.. but in Vaishnavism predominantly it is visishta adwaitam.
Adiyen
Thirukachidaasanudasan