These three banks are “private sector banks”. They’re not “private banks”.
The two terms are often used interchangeably in India, even by leading media outlets as I highlighted in the following tweet.
After mislabelling full-service private sector banks as "private banks" all this while, @EconomicTimes finally gets the real meaning of private banking. pic.twitter.com/T1pcCLu10V
— Ketharaman Swaminathan (@s_ketharaman) March 7, 2018
But just because people use the two terms interchangeably doesn’t make them interchangeable.
Private sector bank is not the same as private bank.
Private Sector Bank means not owned by government. It’s the opposite of Public Sector Bank.
OTOH, Private Bank means the division of a bank that provides wealth management services to High Networth Individuals and Ultra HNIs. These services include estate management, exotic investments, and so on that are NOT offered to regular retail banking customers.
Many – though not all – private sector banks, including the three mentioned in the original question, have private bank businesses.
Technically, even a public sector bank can have a private bank inside it. If I’m not mistaken, a couple of them even do.
Now, coming to the second part of your question, that’s answered easily.
Banks make money if you pay a store with credit card or debit card but spend money if you pay with cash. Revenue sources of the card business include merchant fees, advertising fees, interest charges, etc. Cost sources of the cash business include cash handling charges at branch and ATM for cash disbursement.
Therefore, it’s in a bank’s interest to incent its cardholder customers to spend with cards rather than cash. They not only make money but save money that way – a double whammy of sorts, but in a positive sense.
Cashback is a form of incentive for diverting spends from cash to credit and debit cards.
Ergo, banks offer cashback for purchase of laptop etc. with cards.