Would you say सफ़ेद, गरम, अख़बार, वक़्त, मुज्ररिम, इलज़ाम, ख़त, etc are not Hindi words? Would you rather say श्वेत, ऊष्म, समाचारपत्र, मुजरिम, आरोप, पत्र? Most people use them interchangeably.
For what it's worth, मुजरिम is also Urdu. Not sure about आरोप, but I have a feeling that it's also Urdu, though I could be wrong.
Edit: read any Hindi newspaper these days, at least the online edition, and in fact you will find a lot of common Hindi/Urdu words being replaced with English words written in Devnagri.
Moreover, there are hundreds of dialects of Hindi spoken in India, each with its own nuances and varying degrees of Urdu words.
In addition, there is no denying that Urdu and Hindi have some shared lineage.
Again will refer to the Wikipedia [0] for better coverage of this topic
What else makes a language? Syntax? Hindi and Urdu have almost identical syntax. There are more syntactical differences between Hindi and Punjabi than there are between Hindi and Urdu.
If you want to make your argument rigorous please give an estimate of pure textbook Hindi speakers (people who use avashya instead of zaroor, hriday instead of Dil outside academics and newspapers) and "hindustani" speakers.
My conservative estimate is less than 10M for "pure Hindi" and greater than 300M for regular hindi/Urdu.
This is because virtually every one from North India and Pakistan speaks the so called Hindi/Urdu mixed language. I have also met bhojpuri, awadhi, marwari speakers but I am yet to meet the mythical pure Hindi speaker and I am a really old guy.
Yes there were cultural/academic differences and many different dialects of Hindi. Kabir sounds very different from Ghalib. But for the man on the street there was no difference. At most, you can argue that Urdu is the most popular dialect of Hindi. It wouldn't be correct, but less wrong.
Hyderabadi and Tamil Muslims speak a dialect of Hindi that they call urdu that would make Ghalib spin in his grave.
Differences within Urdu/Hindi are far more tremendous than the Hindi Urdu split. Calling them separate languages is crazy.
It's a cultural standard. Muslims call their language Urdu. Hindus call it Hindi. The language is the same, whatever you call it.
https://en.wiktionary.org/wiki/%E0%A4%9A%E0%A4%BE%E0%A4%AD%E...
Likewise, निश्चित certainly isn't Hindi in this narrow sense. It's a direct borrowing from Sanskrit, not a native, inherited word of the language.
And Hindi (हिन्दी) itself isn't a Hindi word; it's Persian (likewise Hindu).
If we discount direct borrowings from Sanskrit, I guess there would be very little left. Something like Pali, essentially.
So 'fire' in Hindi(/Urdu) is āg (आग), but 'fire' in Sanskrit is agni (अग्नि), which has also been borrowed into Hindi (as the name of the god of fire; or fire in ritual contexts). Now, āg descends from agni, but represents the natural linguistic development into Hindi.
Fire in Pali is 'aggi'. Seems a plausible link between 'agni' and 'aag'.
But you suggested there would be very little left. Yet the core of both Hindi & Urdu is (unsurprisingly) inherited from Vulgar Sanskrit>(some sort of) Prakrit>Apabhramśa>... , so it's rather a lot of vocabulary that's there.
(1) Christopher Shackle & Rupert Snell. 1990. Hindi and Urdu since 1800: a common reader. New Delhi: Heritage Publishers.
[which is actually now freely available online at https://repositories.lib.utexas.edu/handle/2152/46072 (had I known, I would have linked it earlier/higher in the thread, since it's very relevant) its bibliography is also good for further reading]
For the earlier development of modern Indo-Aryan languages from Sanskrit:
(2) Jules Bloch (ed. & translated by A. Master). 1965. Indo-Aryan from the Vedas to modern times. Paris: Adrien-Maisonneuve.
For a good general overall (including history and other things) to Indo-Aryan languages:
(3) Colin P. Masica. 1991. The Indo-Aryan languages. Cambridge: CUP.
In Hindi (modelled on S.K. Chatterjee's The Origin & Development of the Bengali language [written in English!]):
(4) Tiwari, U.N. 1961. हिंदी भाषा का उद्गम और विकास [hindī bhāṣā kā udgam aur vikās]. Prayag, Allahabad: Bharati Bhandar.
Another great (free, online) reference is the Digital Dictionaries of South Asia (including non-Indo-Aryan languages of India as well) at: https://dsal.uchicago.edu/dictionaries/ (perhaps esp. including Turner's A comprehensive dictionary of Indo-Aryan languages: https://dsalsrv04.uchicago.edu/dictionaries/soas/ ).
[Some of the other references can be found 'freely' online in pdf form in the usual 'dark' corners.]