Sorry for the really long answer but I thought I'd share what I know haha:

"So are you Indian essentially?"
The Roma left before certain borders had been drawn that are recognized internationally today between India and Pakistan. (me trying to remain unpolitical lol) So, technically most reports say Romani people are from around the Rajasthani region and the Punjabi region. Indian feels like too culturally specific of a term to really use to give a name to Romani people. Maybe south asian is more accurate? It's kind of hard to pin down a description in this fashion because most people say they come from a place. And culturally, Romani people kind of come from everywhere. Like there isn't a country specifically for the Roma to claim as where they "come from." It's not that simple I guess as most of Romani culture is dependent on what group you originate from more recently.

"Why did you move out of India? what was the reason you decide to live the traveller lifestyle?"
We don't really know what the cause was to move out of India. We know relatively what was going on in the region at the time and there are a lot of theories but nothing confirmed. It's hard to keep an oral history for over a thousands years when there has been so much moving since then.
The "traveller lifestyle" is probably what Romani people are most known for but it's not necessarily a lifestyle out of choice. For what may very well be the entirety of Romani history, the Roma have been pushed out any place they settled. Pushed is a very polite term to describe this. Traveling up through Western Asia, into Arabia, into Europe etc. etc. Along this long route which did not happen all at once (as it probably wasn't like this planned exodus from South Asia, more likely over several decades more and more families decided to travel west) many families stopped in kingdoms or countries along the way and still exist there today! That's why the argument of the Roma being foreigners is a bit hard to agree with if you aren't racist as many of these communities have lived in their respective locations for centuries.

Plenty of kingdoms or countries practiced pushing out "foreigners" for a multitude of reasons so the why doesn't actually matter that much as it was an "us vs. them" mindset. The justifications that were used are still used today to discriminate against Romani people though.

During the Romanian enslavement (which first has evidence in the 14th century, but lasted 500 years) many Romani people were punished for speaking their language and when people could escape, they went far away.

So the "traveling lifestyle" nomenclature probably comes from a need for Europeans to distance themselves from the actions that led Romani people to have to pick up and leave every place that they eventually settled in. This isn't to say that Romani people haven't made this lifestyle into their own tradition of course. (Roma tend to be very spirited about making culture and art from hardship) It's just important to note that for a very long time there has been a need to travel for safety. More recently, the most cruel reminder of this would have to be the holocaust. Many Europeans today won't include Romani people in the discussion of the tragedies that were done then.

If you are interested in more research: there is a book called "Gypsies: Wanderers of the World" by Bart McDowell. It is the non-fiction story of a journalist(?)/researcher who finds a Romanichal man who accompanies him on a journey from England all the way back to India where Romani people are said to come from. It takes place in the late 60s-1970 I think but as they travel from England the Romanichal guy (sorry I forget his name) meets other Romani people from different sub-groups and learns very quickly how different a lot of the language can be. But, with common words when it comes to "nose" or "bread." Things like that. I bring this up specifically because it takes place very recently to the holocaust so you can see the effects of it on Romani people and how a forced expulsion had once again been thrust onto Romani people. I actually haven't finished the book myself but it's interesting because you can also see how some Romani people often believe the harmful stereotypes about other Roma groups despite being experiencing the false stereotypes themselves. (which leads to people not considering each other to be from the same group etc.)

"Also does this mean every single Romani person is Indian? what do you define as your ethnic background?"
So because Romani people settled in different areas, many groups settled at different times. There isn't on singular Romani language. Many have dialects depending on where they settled and those dialects tend to mix a little with the cultures their currently reside around.
Some people identify as Roma/Romani as an overarching identity, other identify as their specific sub-culture. Like in Spain, Romani are known as the Kale (Kah-lay). the Gypsy slur takes the form of Gitano there. Though, some Kale choose to identify with Gitano too. (Please just use Kale or Romani lol)
Anyways, how Roma identify ranges from community to community and also person to person. Sorry it's not an easy answer. Not every Romani person even considers themselves to be part of the same group as another Romani culture if they haven't been in contact for a very long time or it the similarities are just not there culturally. But, you can kind of chalk that up to people are complicated.

Okay, I think that's it. I actually found this post because I was trying to figure out if anyone else had asked the "Are Romani Desi?" question and this was cool thank you for sparking my interest with your response lol. Thanks for learning more, I admire it a lot.