In his PHD thesis, Dazey (1987) discusses these four sampradāyas associated with the four pīṭhas of Daśanāmi sampradāya. I would be quoting that discussion below -
There's an interesting further division of the Order (Daśanāmī) into subgroups called gotra-s and sampradāya-s. Each of the four pīṭhas has a gotra name and a sampradāya name associated with it. All members of the two or three lineages (Sāgara, Vana, etc.) attached to a particular pīṭha said to belong to the gotra and the sampradāya associated with that pīṭha. ~ (Dazey 1987: 452-453)
The sampradāya names, gotras, lineages, etc. of the four āmnāya pīṭhas, can be known in detail from the Mahāmnāya texts, for their source text and translation, refer Clarke (2006:274-286).
The table on Shringeri site which mentions pīṭhas, devatās, kṣetra, etc. follows Mahāmnāyas.
A better (and detailed) chart, based on Mahāmnāyas, is mentioned in Clarke (2006: 118-119) .
Although the
division of the Order into groups called gotra-s and sampradāya-s
seems to be purely formal with no distinctive social function assigned
to them individually by contemporary Daśanāmins, the term gotra itself
may present a clue to the underlying model for the Daśanāmī Order as a
whole..... ~ (Dazey 1987: 452-453)
So, as per Dazey, the four sampradāya names (associated with the four pīthas) don't have any distinctive social function assigned to them by contemporary Daśanāmins. Therefore, these sampradāya names don't exactly denote different sampradāyas (like say Śrī Vaiṣṇavas have Vaṭakalai and Teṉkalai divisions), and this would get clearer from the below explanation.
Although the general meaning and function of sampradāya as a teaching lineage is clear, the rationale behind the actual sampradāya names used in Daśanāmi Order is far from apparent. The names are : Ānandavāra attached to Jyotir maṭha in north, Kīṭavāra attached to Dvārakā in the west, Bhūrivāra attached to Śṛṅgerī in the south, and Bhogavāra attached to Purī in the east. The usual explanation given for these names is that they indicate the peculiar ascetic qualities to be cultivated by the members of the sampradāya.
There are various explanations of these terms given in the oral tradition. Ānandavāra is said to mean those who find ānanda in whatever food they happen to get without begging, or an alternative explanation, those whose ānanda is not derived from worldly pleasures. Those who eat as little as insects (kīṭa) are called Kīṭavāra, or another explanation, the Kīṭavāras are those whose karuṇā extends even to insects (kīṭas). Bhūrivāra is explained as those who renounce gold (bhūri) and other wealth ; or as those who live off the abundant (bhūri) vegetation of the forests. The Bhogavāra-s are said to be those who are particularly indifferent to bhoga.
The etymological history of these terms is obscure, and several theories could be advanced to explain them. For e.g., they may be connected with astrology calculations (kīṭa is the sign scorpio, bhūri can mean many, bhoga can mean part of the ecliptic of the 27 nakṣatras, ānanda means both 16 th muhūrta and a yr in the cycle of Jupiter, and vāra can mean time or moment) or they may refer to group insignias, such as fly whisk or sieve made of hair (vāra can also mean the hair of an animal's tail). Unfortunately, there's little evidence to support these conjectures. What is clear, however, is that the traditional explantions given by contemporary Daśanāmins emphasize the ascetic character of the four non-householder teaching traditions.
~ (Dazey 1987 : 476-477)
Here, Dazey primarily follows Ghurye (1953:97) & Rose (1914:357) in explaining the sampradāya names, but Clarke (2006: 121-122) adds that in Daśanāmī practice these sampradāya names are merely a way of identification of one's lineage.
The origin of the four sampradāya names, Ānandavāra, Bhūrivāra, Bhogavāra and Kīṭavāra, that are given in the texts cited, similarly defies adequate explanation. Most commentators follow Ghurye (1964:86) in explaining, somewhat vaguely, the sampradāya in terms of life-style. However, in Daśanāmī practice, the sampradāya names, as wth the gotra-s, simply confirm to which of the four groups of lineages the initiate belongs, and do not signify a different life-style. The sampradāya names are used by Daśanāmīs as an identificatory title, such as, for example, Mahant Lāl Purī, Bhūrivāra. (It can be seen, according to the Mahāmnāyas, that Bhūrivāra indicates a Purī, Bhāratī or Sarasvatī ; Ānandavāra indicates a Giri, Parvata or Sāgara; etc.)
References
- Clarke, M. (2006). The Daśanāmī Saṁnyāsins: The Integration of Ascetic Lineages into an Order (J. Bronkhorst, Ed.). Brill. https://archive.org/details/dasanamisamnyasi0000clar/page/120/mode/2up?view=theater
- Dazey, W. H. (1987). The Dasanami Order and Monastic Life. (Publication No. 8811826) [Doctoral dissertation, University of California - Santa Barbara]. ProQuest Dissertations & Theses Global. https://www.proquest.com/openview/ea57ab76f4b794b54698b404b35dd9d4/1?pq-origsite=gscholar&cbl=18750&diss=y
- Ghurye, G.S. (1953). Indian Sadhus. Popular Book Depot.
https://archive.org/details/indiansadhus0000ghur/page/96/mode/2up?view=theater
- Rose, H. A. (compiler) (1914). A Glossary of the Tribes and Castes of the Punjab and North-West Frontier Province (Vol. 3). Based on the Census Reports for the Punjab, by Sir Denzil Ibbetson, in 1883, and by E. D. MacLagan, in 1892. Punjab Government Publications. https://archive.org/details/b2901086x_0003/page/356/mode/2up?view=theater