The grammar is equal for all. A sect or a sub sect can't discard the rules of grammar while interpreting a name or a verse written in Sanskrit for their ease to highlight their favorite god. If it is done, then the interpretation can be neglected.
There is a Panini sutra from the ashtadhyayi.
pUrva-padAt saJjJAyAm a-gaH 8.4.3
पूर्व-पदात् सज़्ज्ज़ायाम् अ-गः
Meaning : If a compound is a tag and its former does not end in g ग् (it causes /Natvam in its latter).
In simple words, if a samjna (proper noun) is intended, na-kara gets a ṇa-kara (adesha). i.e., when a specific person is intended by a word, the na- will be replaced by Na if the word doesn't end with g (as in god).
For example, Rāmāyaṇam has a ṇa to denote it is a proper noun. It means the journey of only one Rama not any other rama like Parashu Rama or Balarama.
Whereas, this doesn't apply for a word like Dakshiṇāyana because Dakshina is not a proper noun. It is a common noun which can have many meanings. Dakshina means right, south, starightforwad etc., the word can mean any meaning. This does not happen for the word
When they say Nārāyaṇa, it means only Vishnu and not any other God who is born on water. There is a proper noun i.e., specific person is intended. No other god can be called by that name.
Now, who is the (only) Nārāyaṇa referred to?
In Srimad Bhagavatam skandha 10, chapter 14, there is a eulogy by Lord Brahma where he also gives the definition of the name. I don't think anyone can explain the definition of the word Nārāyana better than Lord Brahma.
14 Are you not Nārāyaṇa? Yes, you are Nārāyaṇa inasmuch as you are the soul of all embodied creatures, who thus form your dwelling place (and nāra means the aggregate of living beings). You are Nārāyana as you are the Lord who preside over and promote life in alt beings, and being the witness to the entire universe, you alone know them all. You are the Nārāyaṇa—the Deity that abide in (and thus are the basis) of all the twenty-four principles evolved out of Nara, as well as waters—your abode for reposing, which is the evolute of Nara who is also your part. Even your reposing on water is also not true, but your māyā, as your form is indiscernible.
So, Brahma gives several definitions already. How did he give those meanings?
There are different etymologies for the word Nāra
nārāṇāṃ ayanatvāccha nārāyaṇa iti smṛtaḥ
nāra śabdāna jīvānāṃ samūhaḥ procyate budhaiḥ
Nāra - aggregate of all beings. ayanam - Hence who dwells in all bodies as their souls.
The Vishnu Gayatri mantra is as follows: (From shruti):
Nārāyanāya vidmahe vāsudevāya Dheemahi Tannah Vishnu prachodayat
In the above Gayatri mantra, the names Nārāyana, Vāsudeva and Vishnu are equated. According to the above etymology, Nārāyaṇa means who dwells in all bodies as their souls. Vasudeva also means the same.
He is called Vasudeva in consequence of his enveloping all creatures with the screen of illusion, or of his glorious splendour, or of his being the support and resting-place of the gods. He is called Vishnu because of his all-pervading nature. [From What is the meaning of Brahman & Vasudeva?]. So, the name Nārāyaṇa refers to Vishnu and not other god.
Nārasya ayanam pravrittih - The source of the promoter of all the aggregates of Jivas.
Nāram ayase - You know the aggregate of all beings. ay means to know.
Nāra - which is produced from Nara
Nāra - who dwells in or reposes in water
teṣāṃ ayana bhutatvān nārāyaṇa iti smṛtaḥ |
nāro narāṇāṃ saṃghatasa tasyāhaṃ ayanaṃ gatiḥ |
tenāsmi munibhira nityaṃ nārāyaṇa iti smṛtaḥ ||
cetanā cetanaḥ sarvaṃ viṣṇor yada vyatirincite |
nāram tadayanam cedaṃ yasya nārāyaṇāstu saḥ ||
āpo nārā iti proktā āpo vai narasūnavaḥ |
tā yadasyāyanaṃ pūrvaṃ tena nārāyaṇaḥ smṛtaḥ || Manusmriti 1.10 ||
These are the couplets which were the basis of those above etymologies. These are quoted by various commentators on the above specific verse from Srimad Bhagavatam.
Because all the sects and commentators adhere to Panini's rules of grammar while interpreting and commenting names of god, we should not limit this meaning to a single sect. All accept the meaning equating to Vishnu.