Why Banks give more discounts/offers on Credit Card as compared to Debit Cards? Whether a credit or debit card is used to make a payment at a merchant, there will be outflow of money from the bank. What extra benefit do the bank get from promoting credit more than debit card?
Is it just the obvious answer that by promoting credit card, there is a chance that user will default on the payment and allow bank to charge interest?
On the downside, by paying through credit card, the user still has money in her savings account and so from bank has to maintain higher reserve ratio with the central bank.
Finally, does it have something to do with the fee bank charges with merchants? Is the fee higher for use of credit cards as compared to debit cards?
Yes of course you are right about both of your points. The fee charging the credit card POS transaction is about 2% to 3% depending on the company. The fee charging to debit card transaction is about flat 25 cents or lower.
For online transaction, the credit card fee can be as high as 4%, and the debit card fee can be negligible. Banks make more money on credit card.
So, to answer your question:

Credit department usually have more money to make promotions.
Average people (in US) have almost zero savings so they cannot use their debit card very often. But everyone has a credit card.
Of course, credit card can default, and the bank can legally charge hefty interests.

And as mentioned by Brian, competition is an important reason.