That goes against what I heard from my Indian friends. I was told that public schools are actually more prestigious than private schools, they are more rigorous apparently?
Probably a carryover in terminology from Britain, where extremely prestigious schools like Eton are called "public schools" for historical reasons.
In the UK public and private referred to who could access the schools (assuming they had the money), whereas in the US they refer to how the schools are funded.
Now the UK has a kludge. The former private schools don't really exist, they've mainly changed into independent fee-paying public schools not controlled or funded by the state.
In the meantime (i.e. centuries ago), the state started funding schools, including religiously selective schools and has been increasingly giving more independence to the schools it funds.
No, 'public school' would be understood to refer only to the most elite of private schools in the UK, such as Eton, Harrow, Winchester, etc. A run-of-the-mill private school would not be considered to be a public school.
They are, but they are still private schools, not government schools. They are based in British public schools, where the word 'public' refers to the fact that students of any religion could historically be admitted to them, not that they are state-owned or state-controlled.
Sometimes private schools can be shittier than the government run schools (eg. in the town my family is from in Himachal) but in other places the private schools will at least not have absentee teachers (a common issue in much poorer regions of India).
P.S. the public school moniker for some Indian private schools is a holdover from the British era. Most modern private schools don't call themselves "public" anymore - they prefer being called "International Schools".
Public schools are a joke and you would usually find kids mocking anyone slightly unsophisticated as “from a government (public) school”