The subjects of the Mughals adopted their religion, influenced by their culture, famous Mughal monuments like Taj Mahal dot the subcontinent, the legacy of sufism and emergence of Sikhism, all of these are their traces found in the substance of what it means to be Desi. Not all of these are shared by all people under the term Desi, for example the history of a person from Tamil Nadu will differ markedly from the history and view of someone from Punjab. But they are still part of the overarching sense of the Subcontinent. The Mughals and previous Sultans brought a new civilization that fused with the identity of the region over time.
Conversely, despite the British way being heavily influential in the Subcontinent and having had the most recent influence on the makeup of the countries there, nobody will mistake the Buckingham palace as having the same legacy as that of India. The British colonial era was clear in transferring wealth away from the region and the British did not consider India native to them. The Mughals came as conquerers, yes, but they became integrated into the culture they conquered. They also brought wealth to the region. Did the British Monarchs ever marry into the culture of their subjects like the Mughals did?
For the British, India was a profitable "Crown Jewel" but the mission was not to find the new land as a home. See how their presence in the region began as a trading company.
Contrast with the US where Native Americans were forced into reservations and their culture and language has little influence on most of the inhabitants of the US today. But the idea of the US also departs from a lot of ideas about nation, it is its own beast.
You’re the one doing that. But rolling back the clock 300 years before the British, but not 500 years before that before the Mughals and Sultans is quite indefensible.
> By indigenous I define it as the people of the region who settled there and became native to the land.
By that logic Americans are indigenous to the US but that’s not how we usually use the term. And Mughals weren’t that either. They were a foreign elite that conquered the subcontinent by force. Your average desi has very little Mughal ancestry.
> It’s a contrastive term for distinguishing from the British colonizers.
It’s a paper thin distinction.
> To move forward as a society one needs investment in native institutions
The irony is that Europe itself did quite the opposite. Their society has little connection to that of their Germanic ancestors. Instead it’s a synthesis of a middle eastern religion and Roman/Greek civil society and law.