- So let me share two aspects for your question: One from the Same Purana you quoted, second from the Veda, wherein Brahma gives names to Shiva.
- Here is a link to help you read Chapter VIII-7 of Bhagavad Purana. I am attaching the image from this chapter that answers your question. I would request that you read this chapter to get a complete idea. ( https://archive.org/details/bhagavatapuranaeng031955ocrmotilalbanasirdass/page/n151/mode/2up)
Now coming to the 2nd part of our answer from Vedas:
The birth of Rudra and Gāyatrī
Now, let us look at a metaphysical event of Prajāpati’s Yajñá from Sukla Yajur Vedā Śatapatha Brāhmaṇa Kanda 6 Brahmāṇa 3 (6.1.3.10-17):
In the beginning, Prajāpati was alone. He desired “may I multiply”. Upon His practicing austerity, there emerged fumes called Tapo:dhuma, from which came water. Water wished to flourish, upon austerity came foam; foam wished to flourish and upon austerity came clay. In this way each desired to be more and so performed austerity in the following order: water > foam > clay > sand > pebbles > stone > metal ore > gold > Akṣarā (imperishable syllables/sounds). Since this bloomed into eight-folds, the 8-syllable Gāyatrī emerge. All these became the foundation for Bhumi, which spread out (prath), hence became Pṛthvī. All continued their austerity for a Sāṃvatsara, meaning one year, and upon its dawn (Uṣas/Auṣasī) arose a child, meaning the union of Sāṃvatsara and Uṣas gave birth to a child. Upon birth, this child cried (arodīt); Prajāpati held this child and said “my son you are Rudra” (the first roar/sound). This is why Rudra is the primordial roar. Hence, please note that Rudra didn’t come as the boy, this boy was called Rudra. This boy could not be contained and wished to encompass everything and so said, “with my name I have to become greater” and asked for another name.
Prajāpati said “you are Sarva” and so the water became him, the boy grew in greatness and kept asking for more names.
He was named Paśūpati, from him emerged flora/fauna.
He was named Ugra (the fierce one), Vayu/Wind/Air became him.
He was named Aśani, lightning became him and so did thunder.
He was named Bhava, the rain became him and the Divinity of rain is Parjanya.
He was named Mahādeva, the highest Divinity, and the moon (Soma) became him; hence He, Mahādeva with a moon on his head.
He was named īśhana (the authority/the ruler), the Sun became him.
Please Note: Rudra here is the Agni itself. And Aśani the thunder becomes the power of Indra. Mahabharatam 7.175.96 specifies the weapon/missile of Rudra as “aṣṭacakrāṃ mahāghorāmaśaniṃ rudranirmitām“. The Samhita portion of Atharva Vedā 15th Kanda 18th Prayāya, known as Vrātyas Suktam state 7 of the above title, and the only name missing is Aśani, as if Aśani is centered among the 7. Kṛṣṇa Yajur Vedā Taittirīya Saṃhitā 4.5.5 use the same names except for Aśani, so these titles are not new, nor limited to Brahmāṇa section. When this nameless boy entered all aspects of creation and became creation itself. A much detailed event can be seen in the event of Śatarudrīya of Sukla Yajur Veda. Finally, Manu is not there in this list nor in the list of Śatarudrīya.
In the context of the Yajñá, which was performed for one full year (Sāṃvatsara) and at the end of which arose new dawn (Uṣas), from this fire-altar/Agni came all 8 forms of Rudra. These 8 forms are called the Astamurti Tatva. Together, the Agni/fire-altar, the boy, and the 8 forms of Rudra are the 3-fold state of Agni:homa. These 8 forms are the eight syllables of Gāyatrī. Since Gāyatrī is sung in 3 padas (steps) the meter is 8×3=24. In the prominent Śatarudrīya Yajñá, Prajāpati sings Sāmans like Gāyatra using the Gāyatri meter. He does so while circling around the altar three times (pradakṣiṇa) with the altar being on his right, indicating its highest stature. Among the 70 Svayambhu Liṅgas Mritakeśwara takes the 35th position in the list and is called Gāyatrī. Mahabharatam Dharma Parva Section 285 (Jaya section) concurs with this by saying “Thou art the beginning and thou art the end of the Vedas, thou art the Gāyatri and thou art OM.” If you notice this statement, its also concerns with Bhagavad Puranam slokas in the pic.
Now we continue into the next chapter, Vājasaneyi Saṃhitā 8.2.3.6 Brahmāṇa 1. Since the child became free (Swa) and became all (roar/sound, water, wind, living beings, lightning, rain, Soma/moon & Sun), his father the Prajāpati searched for him in Agni and in Paśūs (Puruṣa, horse, bull, ram, he-goat) but the boy was not found (Apaśyat). There was only Agni, so Prajāpati realized that everything was Agnihotra. This event of the Yajñá explains the raw, unmanifested nature of Rudra who can exist in all, the innate self of all. His three eyes are Soma, Agni, and Surya and He wears a crescent moon on His head. He is “Maha” of Devas and He is īśha (the authority) and He is Swa (free/sovereign); hence He is Maheśwara (Maha + īśha + Swa), the highest of names (paranama VS10.2) and “mahó mahī́ṁ ..gr̥ṇīmási tveṣáṁ rudrásya nā́ma” RV2.33.8.
For detailed research on Rudra, you can search in Google for "Sanatanadhara Rudra Shiva across Vedas and Itihasa"