The three Entities are as follows.:
- Vrisha/Vrishava is Lord Vishnu himself.
During the Tripura Dahan episode, Lord Vishnu assumed the form of a bull to stabilize the chariot made of Sun, Moon, Seshnaag, etc which couldn't handle the the weight of Bhagawan Shiva.
Sri Linga Purana 1.72.:
28-31. Eulogised by the sages and saluted by the bards the shining splendid lord, the bestower of boons occupied the chariot glancing at the charioteer. The groups of Apsarases, skilled in dancing, danced in his presence (to honour him). When he got into the chariot, evolved out of the different material, the horse originating from the Vedas fell headlong over the earth. The lord Dharaṇīdhara (uplifter of the earth i.e., Viṣṇu), assuming the form of a leading bull lifted up the chariot for a while and tried to stabilise it. But at the next moment even that leading bull slipped down to the earth on his knees.
Then after the destruction of Tripuras, in the same chapter only, Lord Vishnu asked the boon to be Bhagawan Shiva's Vahan. Which he granted happily.
- Lord Viṣṇu also bowed down to Maheśvara. Joining his palms together in reverence he said thus to the three-eyed lord accompanied by Umā.
174-175. “O lord, be pleased with me. O lord of Devas, obeisance be to you. I perpetually desire to be your vehicle. I wish for your devotion as well as my efficiency to bear you. O Śankara, the bestower of boons, I wish for omniscience and all-pervasiveness.
Sūta said:
- On hearing their submission Parameśvara, Mahādeva, Bhava engaged them respectively in charioteership and the position of vehicle.
Again, Sri Nandi is also called the bull vehicle of Bhagawan Shiva.
Sri Shiva Purana 2.5.54.:
- In order to help Bāṇa, lord Rudra accompanied by his son and the Pramathas rode on the bull Nandin and arrived there to fight.
Bull represents Dharma.
ŚB 1.16.18.:
The personality of religious principles, Dharma, was wandering about in the form of a bull. And he met the personality of earth in the form of a cow who appeared to grieve like a mother who had lost her child.
So, Bhagawan Mahadeva rides Dharma itself.
Just as Mata Savitri and Mata Gayatri in some scriptures are described as 2 different persons and in other places as just Mata Saraswati. Similarly, Lord Nandi/Nandishwara (Lord/Bestower of Happiness) and Vrishava are sometimes showed as 2 different entities.
MBH 13:14.:
I then behold the illustrious Sthanu or Mahesvara seated on the back of his bull, of blessed and agreeable appearance and looking like a smokeless fire. And the great god was accompanied by Parvati of faultless features.
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In the front of Mahadeva I beheld Nandi standing armed with his Sula and looking like a second Sankara (for prowess and energy). . .
- Bhringi/Bhringisha is the Asura Andhaka.
Sri Skanda Purana 5.1.38.:
Īśvara said:
86-90. Well! Well! O Dānava of great intrinsic strength, request for a boon. I am the donor and you are the suppliant. I shall grant whatever is desired.
Andhaka said:
O Lord of Devas, if you are pleased, if a boon is to be granted to me, then I should be made one on a par with yourself. No other boon. I shall be Bhasmī (one smeared with ash), Jaṭī (one having matted hairs), Trinetrī (having three eyes), Triśūlī (having a trident), Caturbhuja (having four arms), having a tiger hide as the upper garment and serpents for Yajñopavīta. O Maheśvara, if you are pleased, I shall desire these.
Īśvara said:
I shall grant you the boon as requested by you, O sinless one. Dear son, stationed among my Gaṇas you shall become Bhṛngīśa.
- The scriptural references for Lord Chandesha, can found on this site.
I hope this clarifies all your queries. Prd..