Do people in India really use these months, or do they use different names for them from their own calendars?
There is a Hindu Calendar which was also divided up into 12 months, but things were a little complicated. As I understand it, some regions had months starting on the new moon, while others started them on the full moon. You can read all the details here: https://en.wikipedia.org/wiki/Hindu_calendar
So yes, to answer your question in modern India they generally use the anglicised names.
Just to add to this, there are actually multiple traditional Hindu calendars. Most of them are lunisolar but some use solar months (based on the sidereal zodiac) and others use lunar months (based on the phases of the moon). Then, as you say, is the difference between starting a month from the new moon and the full moon. There is also disagreement about the day a year starts (ie, which month is the first month of the year). Even if two calendars agree on all these counts, they may still differ in what year it is because they started counting from different years (the Vikram and Saka eras being the two most popular). The calendar you use depends on the region you come from and your family tradition. They are now mostly used for religious purposes and for determining dates of festivals and observances.
There is also a calendar that was created by the government in the 1950s called the 'Indian national calendar' that uses months with the same names as most traditional calendars (चैत्र, वैशाख, ज्येष्ठ, आषाढ़, श्रावण/सावन, भाद्रपद, आश्विन, कार्तिक, मार्गशीर्ष, पौष, माघ, फाल्गुन in Hindi). It was meant to replace traditional calendars and introduce some uniformity. However, since people were already using the Gregorian calendars for most secular purposes and were loathe to give up on their traditional calendar for one they were unfamiliar with, the adoption of this calendar never gained any traction though some government departments still insist on using it.
You can now see how using the names of the months from the Hindu calendars to refer to the months from the Gregorian calendar may introduce a lot of unnecessary confusion. So, everyone just sticks to the English names.