Here is a partial answer:
Varuna, Agneya, Vayabya etc - these forms of baths are mentioned in various scriptures. My reference here is Pararshara Smriti's 12th chapter.
SnAnAni pancha punyAni kirtitAni manishibhihi |
Agneyam vArunam brAhmam vAyabyam divyameva cha ||
- The learned have declared five kinds of bathing to purificatory : the Agneya or fiery bathing, the Varuna or the watery bathing, the
Brahma or Vedic bathing, the VAyavya or windy bathing, and the Divya
or the celestial bathing
10 — 11. The Agneya bathing consists in bathing with ashes ; the
Varuna is plunging into the water ; the Brahma consists in reciting
the verse ' apohishta, &c. ;' the Vayavya is bathing with the dust
raised by the hoof of a cow ; but the celestial bathing takes place
when one is drenched with, rain-water, the rain coming down while the
sun is shining. This kind of bathing is of equal efficacy with the
bathing in the river Ganga.
So, this scripture is talking about five kinds of baths.
This is the daily mandatory bath that one should perform everyday at early morning (preferably before the sun rises).
This is an occasional bath which is done for a particular reason for e.g. Grahana baths (baths after eclipses), SmkrAnti baths or baths when a particular impurity (Asaucha) has ended.
This is a bath done keeping a specific purpose in mind. Examples will be baths taken in holy Tirthas/rivers.
These three kinds of baths are mentioned in Daksha Smriti for example:
Bathing has been spoken of as being threefold, viz. Nitya (daily),
Naimitika (obligatory, as in the solar or lunar eclipse), and Kamya
(having an end, such as the attainment of the celestial region in