Narayana
The Mahamrityunjaya Mantra Has been used in the Vaishnava agamas in the praise of Lord Vishnu, for instance
~ Shesha Samhita, chapter 46
~ rishyasringa samhita 2.7.2
While the Vimanarcha kalpa declares it to be a vaishnava mantra as "रुद्रमन्य' मितिबन्धयेत् 'त्र्यंबक' (ऋ. सं. ५-४-३०) मितियुगं, लांगलंच, वैष्णवं मन्त्रमुच्चार्यतां भुवं"
सप्तवारं प्रतिसर'मस्त्रमन्त्रे'ण मन्त्रितम्।`त्र्यम्बकेन' ऋचालिप्य चन्दनक्षोदवारिणा ।। 14.14 ।। ~ bharagav Tantra 14.14
पञ्चगव्यं शिवञ्चेशमव्यक्त' मिति पूजयेत्।
जपित्वा विष्णुसूक्तञ्च 'रुद्रमन्यं-त्र्यम्बकम्' ।। ~ kriyadhikar 33.46
Obj : The Anukramika names Rudra as the deity of the sukta, while The panchabrahman mantras Clearly say that Rudra can Grant moksha
Resp : It is not necessary that the deity Mentioned in the Anukramika is the same you Have to talk about, because Shastras say "रुद्रो रोदयते तस्मात्ब्रह्मा बृंहणकर्मणा" (Bhrigu.sam.36.221) i.e Visnu Makes Us Weep hence he is Rudra, "ब्रह्मरुद्रादिदेवानामगम्या सूक्ष्मतावधिः" (Sankarshan.Sam.2.1.21) i.e Visnu is denoted by all names of Rudra, brahma etc. "रोगनाशो भवेद्रुद्रो" (Vridh.Har.smriti) i.e Visnu is called Rudra, since he dispels all diseases And so on. Further, Even the panchabrahman mantras as a whole is not an issue for us, since multiple shatsras attribute the usuage of Panchabrahman mantras in worship of Visnu, which includes smriti shastras as well. For instance "ईशानेन तु मंत्रेण शिरस्येव विनिक्षिपेत् । तत आदायतद्भस्म मुखेतत्पुरुषेण तु ॥२४।। अघोराख्येन हृदये ततस्तद्भसितं क्षिपेत् । सद्योजाताभिधानेन भस्मपातद्वये क्षिपेत् ॥२५॥....स्मरन्नारायणं तिष्ठंद्यावर्द्धल्यावृतं पुनः" (Bharadwaj.smr.5).
Moreover, as per Kriya kairava agama, Rama gita of skanda puran, Narayana tapniye etc, Visnu is the Essence of all five faces of sadasiva, hence all Mantras of the panchabrahman must actually denote Nārāyaņa only.
The same goes of Other rudra suktas, Sri rudram etc as well. The shastras very well talk of Visnu being the devata and beholder of these Hymns, which I will address in a separate answer, in a detailed fashion.
Rest, this explaination was from the POV of a vaishnava, you are free to disagree with it ofc and the objective was Not to decide who gives moksha or not, but the deity of the mantra in question, but Before levelling objections, one must be thorough with the Facts before hand.
Obj : The Term pinaki is used, Which refers to Siva alone
Resp : Not so. The Term pinaki has been used by Not only shastras like narada puran, nrisimha tpaniye, Nrsimha puran etc to denote Narayana, but even Acharaya shankar has Done the same in his bhasyam to NTU. Therefore, this is not a Nirvaksha linga for Lord Siva.
Obj : Krishna himself explains and hails Siva as Tryambaka in Harivansha Puran
Resp : And why did he do so, Has not only been explained by Lord Siva himself in Harivansha 3.88, but also by several other shastra vakyas.
Śiva undergoes a realisation just like prahalada in Visnu puran and mentions that since he is an expansion of Visnu, he too is the Lord of all. This is the reason which he recounts for him also having All names of Visnu, one of them being Tryambka as well. Hence it has been said :
ऋग्यजुस्सामार्थर्वभिर्मन्तैः वैष्णवैः देवं संस्तूय नमोऽन्तैर्नामभिः प्रणमेत्... 11
Vişnu is praised in the mantas of Rg, Yajur, Sāma, and Atharvavedas in His infinite names.~ Vaikhānasa Grhyasūtra 3.13.4
And
सर्वेषामेव देवानामनादिः पुरुषोत्तमः 11
Purusottama alone is the beginning of the names of all the deities.
~ Vrddha Hārītasmrti 1.56
Obj : Not giving any other pramana outsides Shrutis as they are svatah pramana when there is no challenge in interpretation.
Resp : This is not a new statement which i have heard ; often people with no understanding of shastras and with an inbuilt negative notion about Purnas, Itihasa, smritis, agamas etc Often end up making such statements, whereas the Truth is, Shrutis are to Be interpreted in the light of Vedangas, smritis, Purans, itihas and agamas. Srutis are Svataha pramana is fine, but Why should one agree with the meaning which you are giving? From where did you get to know that Rudra is clad with Skin and is pinaki? Obv you saw that in Other texts and you are applying the same here. Hence, this logic is inappropriate.
Obj : These upadhis are clear on Shiva/Rudra. Rudra is the Devata associated with medicine and healing in Rigveda especially with healing and protecting cattle. And Rudra is the one clad in skin with Pinaka residing on the mountains (if only wielding Pinaka is not enough to conclude). In Tryambaka Homa (SB 2.6.2), the extremely fierce form of Rudra is also mentioned and satisfied that may protect the Yajaman from his fury on his descendants and cattle (also corroborated with mantras of similar nature in Rigveda addressed to Rudra).
Resp : Nrisimha tapniye Upanishad, Mahabharata etc mentions the Lord Residing on the mountain top. So the assertion is quashed once again. Further, not only Purans, but even Upanishads like Nrisimha tapniye Praise Vishnu with the mantras of Rudra sukta. Even purvacharyas like adi Shankaracharya has Quoted The rudra sukta in his bhasyam to Visnu sahasranama, saying that Vishnu is the one who cures. Hence, neither the Fury Form of Rudra, nor his Characteristics nor his sukta, is exclusive to him.
Obj : Pancharatras are sectrain texts and not accepted as pramanas
Resp : when all the shastras, starting from Purans, Itihasa, Vedas and smritis hail panchartaras, when Even the Abhedvadi acharayas like Shankaracharya, have accepted the Rituals, stutis and practices of pancharatra, so on what basis is this statement made?
Obj : An equal number of completely counter and opposite assertions can be thrown from one side to the other that kills the debate.
Resp : That's not going to happen, because Saiva Shastras don't even Know any murty of Visnu beyond the Karaneswara, who is the Manifestation of sadyojata Mukha of Sadasiva, while vaishnava sastras elaborate clearly upon the entire Hierarchy of Saiva murtys, starting from Paramsiva himself. Moreover, the Vigraha of Siva itself is a result of Sayujya with Sankarshan javalamali Nrisimha, hence all of the mantras dedicated to Rudra, will obviously be the praise of Nārāyaņa only.
