I think, there are already 3 answers which shared the story of the mutual fight between Lord Brahma and Lord Vishnu. So, let me answer in a different way.
Bhagawan Shiva's Agni Lingam in Vedas and Itihasa.:
“sa eṣa rudra bhaktaś ca keśavo rudra saṃbhavaḥ | sarvabhūtabhavaṃ jñātvā liṅge ‘rcayati yaḥ prabhum | tasminn abhyadhikāṃ prītiṃ karoti vṛṣabhadhvajaḥ |” (MBH 7:172:89-90).
“Kesava is that devoted worshipper of Rudra who has sprung from Rudra himself. Kesava always worships the Lord Siva, regarding his Linga emblem to be the origin of the universe. The God having the bull for his mark cherisheth greater regard for Kesava”.
“vadantyagnĩ mahādēvaṁ tathā sthāṇũ mahēśvaram |
ēkākṣaṁ tryambakaṁ caiva viśvarūpaṁ śivaṁ tathā ||” (MBH 13:161:2).
“The Rishis describe Mahadeva as Agni, and Sthanu, and Maheswara; as one-eyed, and three-eyed, of universal form, and Siva or highly auspicious”.
Here lord shiva is called both ‘agni’ and ‘sthanu’
Etymologically ‘sthāṇu’ means – ‘pillar’, ‘immovable’, ‘fixed’, ‘pole’, ‘trunk’, ‘stationary’. It has the same origins as that of the word ‘sthūṇa’ which etymologically means, – ‘pillar’, ‘post’.
The name ‘sthāṇu’ thus means – “he who is a pillar which cannot move because of having filled everything by himself and there is no place left outside him to pervade further”
So he can be described as an agni piller or agni lingam.
“mahat pūrve sthito yacca prāṇotpattisthitasca yat ।
sthitaliṅgaśca yannityaṃ tasmāt sthāṇuriti smṛtaḥ ॥” (MBH 07:202:133).
“And since he is great and ancient and is the source of life and of its continuance, and since his linga-emblem is fixed (motionless) and is everlasting, he is for that reason called Sthanu”.
“vayā idaghne aghnayaste anye tve viśve amṛtā mādayante |
vaiśvānara nābhirasi kṣitīnāṃ sthūṇeva janānupamid yayantha ||” (Rig Veda 1:59:01).
“The other fires are, verily, thy branches; the Immortals all rejoice in thee, O Agni. Centre art thou, Vaiśvānara, of the people, sustaining men like a deep-founded pillar”.
“jvalaliṅgāya namaḥ |” (Taittiriya Aranyaka 10:16:2).
“Salutations to Him (Rudra), He, who has the form of the Agni Linga”.
This Jwala-Linga is termed as ‘Skambha’ in Atharva Veda primarily. Hence entire creation is said to have sprung from his ‘Jwala-linga (Skambha)’ form of Mahadeva as per the below verse from Athatva Veda’s Skambha Suktam.
“skambhó dādhāra dyā́vāpr̥thivī́ ubhé imé skambhó dādhārorv àntárikṣam
skambhó dādhāra pradíśaḥ ṣáḍ urvī́ḥ skambhá idáṃ víśvaṃ bhúvanam ā́ viveśa |” (Atharva Veda X:7:35).
“Skambha set fast these two, the earth and heaven, Skambha maintained the ample air (Antariksham) between them. Skambha established the six spacious regions: this whole world Skambha entered and pervaded”.
“yásya tráyastriṃśad devā́ áṅge sárve samā́hitāḥ |
skambháṃ táṃ brūhi katamáḥ svid evá sáḥ |” (Atharva Veda X:7:13).
“Who out of many, tell me, is that Skambha He in whose body are contained all three-and-thirty Deities.”
“namō bṛhatē ca |” (Yajurveda IV:5:5:i).
“Salutations to Rudra who is ‘bṛhat’ i.e., brahman”.
“atha kasmāduchyatē anantō yasmāduchchāryamāṇa ēva |
tiryagūrdhvamadhastāchchāsyāntō nopalabhyatē tasmāduchyatē anantaḥ ||” (Atharvasiras Upanishad 4:04).
“Next, why is He (Rudra) called ‘ananta’? Because, his end is available neither at the top nor at bottom, nor at right nor at left – therefore He is called ‘ananta (endless)’”.
We have seen from the above verses that lord shiva is the agni lingam which appeared in front of lord Brahma and lord vishnu .
Now, why this present kalpa is called as Sveta-Varaha kalpa.?
Shiva Purana clearly says that Lord Vishnu assumed the form of a white-boar (Svetavaraha) as evident from below verses.
“nārāyaṇōpi viṣvātmāsuṣwētōhyabhavattadā |
daṣayōjanavistīrṇēṣatayōjanamāyataṁ ||” (Shiva Purana 2:07:56).
“And the soul of the universe, viz. narayana assumed the form of a white complexioned boar spreading hundred yōjanā-s in height and ten ‘yōjanā’ in girth”.
“mēruparvatavarṣamāṇaṁ gauratīkṣaṇōgradaṁṣhtriṇaṁ |
hrasvapādaṁ vichitrāngaṁjaitraṁ dr̥dhamanaupamaṁ |
vārāhākāramāsthāya gatavāṁstadadhau javāṭ ||” (Shiva Purana 2:07:57-58).
“Having his chin as like as the meru mountain shining with white hue and having sharp-pointed teeth, resembling the radiance of the form of the sun which is seen at the time of destruction of the universe, having a long nose, creating huge noise, having medium feet, and strange looking parts, stern, victorious; assuming such an incomparable form of the boar, he with an excellent speed, moved downwards”.
This is why Shiva Purana clearly says that because Vishnu assumed the form of a white boar, this current kalpa is named after him as “Sveta-Varaha Kalpa”.
“ēvaṁvarṣasahasraṁ cha charanviṣṇuradhōgataḥ |
tathāprabhr̥ti lōkēṣhu ṣvētavārāhasaṁgyakaḥ ||” (Shiva Purana 2:07:59).
“This way for thousands of years, Vishnu travelled down and down. (In remembrance of that incident) from that day, that Kalpa was famed as ṣvētavārāha kalpa”.
Now some people can say no this is incorrect the Sveta varaha kalpa is because of lord vishnu's Varaha avatar.
So my answer to them is nope you are absolutely wrong because the varaha avatar of lord vishnu was black in colour not white.
There is a verse in taittiriya Aranyaka of Yajurveda which says that Earth was lifted by the Boar of dark complexion which had a hundred hands.
“bhūmirdhēnurdharaṇī lōkadhāriṇī |
uddhṛtāsi varāhēṇa kṛṣṇēna śatabāhunā ||” (Taittiriya Aranyaka 10.1.8.38).
“The earth is the giver of happiness like the milch cow, the sustainer of life and support for all living beings. Thou wert raised up by the boar, by the dark one, having hundred hands”.
See, the dark one which means black obviously.
According to Harivamsa Parva of Mahabharata, Lord Vishnu assumed the form of a gigantic fierce boar of the complexion of Dark-Blue-Rain-Cloud.
“nīlamēghapratīkāshaṁ meghastanitaniḥsvanam |
mahāgireḥ saṁhananaṁ shvetadīptōgradāṁṣṭriṇaṁ ||” (Harivamsa 3:34:31).
“Having a colour, similar to that of a dark blue rain cloud, with a sound similar to the thunder of a rain cloud, with a body similar to that of a huge mountain, with his fang shining like the white island”.
“pīnōnnatakaṭīdēshaṁ vr̥ṣalakṣa-ṇapūjitaṁ |
rūpamāsthāya vipulaṁ vārāhamamitaṁ hariḥ ||” (Harivamsa 3:34:33).*
“Having a large and high hip, typical of a bull worthy of worship, hari (viShNu) took up a vast and unbound form of the great boar (varAha)”.
“rarāja śaṅkhacakrābhyā̃ tābhyāmasurasūdanaḥ |
sūryacandramasōrmadhyē yathā nīlapayōdharaḥ ||” (Harivamsa 3:39:14).
“The slayer of asuras, sparkled with the conch and chakra (wheel) like a dark blue rain bearing cloud sparkles between the sun and the moon”.
Vishnu Purana says Vishnu’s Boar form was of ‘dark’ complexion.
“tataḥ samutkṣipya dharāṁ svadaṁṣtrayā mahāvarāhaḥ sphutapadmalōchanaḥ |
rasātalādutpalapatrasannibhaḥ samutthitō nīla ivāchalō mahān ||” (Vishnu Purana 1:04:26).
“The mighty boar, whose eyes were like the lotus, and whose body, vast as the Níla mountain, was of the dark colour of the lotus leaves, uplifted upon his ample tusks the earth from the lowest regions”.
Then here also Vishnu’s Varaha form’s complexion is narrated to be of Dark-Blue-Rain-Cloud kind of color.
[Srimad Bhagawatam (SB 3:13:33) mentions the colour of the boar as “tāmala-nīlaṁ |” which means, “Of the bluish colour as like as the blue coloured Tamala tree”,
And Padma Purana doesn't state the colour of the varaha avatar.
As, for the Lingo-Bhava story its mentioned in many puranas like ,Shiva, Vayu, Linga, Skanda, Brahmanda, Padma, etc etc..
Now, Why Lord Brahma and Lord Vishnu could not identify each other?
“maayaa.n tu prakR^iti.n vidyaanmaayina.n cha maheshvaram.h | tasyavayavabhuutaistu vyaapta.n sarvamida.n jagat.h |” (Swetasvatara Upanishad 4:10).
“Know then Prakriti (nature) is Mâyâ (art/illusion), and the Maheshwara (Shiva) is the Mâyin (maker); the whole world is filled with what are his members (forms)”.
Why Brahma, Vishnu, Rudra takes birth from each other in different kalpas or in different puranas.?
Vayu said, I shall narrate how Rudra was born and how Brahma and Vishnu were born from each other. The three are Atmans as causes of born of Mahesvara; they are cause of creation, sustenance and annihilation of the Universe consisting of mobile and immobile beings. Endowed with great qualities they are sanctified by great Lord. Presided over by his Shakti, they can perform their activities. Brahma can create, Vishnu can protect and Rudra can annihilate. But they rivaled with one other. Desirous of excelling one other they propitiated their father the Supreme Lord by means of penance. Attaining all around favor of Lord at outset in Previous Kalpa, Rudra created Brahma and Vishnu. In another Kalpa, Brahma created Vishnu and Rudra. In another Kalpa, Vishnu created Brahma and Rudra. Thus in different Kalpas, Brahma, Vishnu and Rudra desiring mutual benefit are born of one another. Based on the events in their respective Kalpas, their process in being the cause of origin of one other is extolled by Sages. (Chapter 13, Vayavivasamhita, Shiva Purana)
I hope this clarifies all your queries. Prd..