What is the ruling on the stipulated increase for delaying debt repayment, and is selling in installments with an increase, called "interest," considered usury?

The scenario mentioned in the question: "That you stipulate to the borrower that he must repay the loan on such-and-such a day, and if he delays, he is obliged to pay back more than he borrowed," is the very essence of forbidden usury (riba), which the Quran has prohibited. It is absolutely impermissible for a Muslim to engage in it.

Borrowing from a bank and purchasing with interest is also forbidden.

Whoever acquires wealth through a forbidden act must repent to Allah Almighty and dispose of the equivalent of this wealth by spending it on charitable causes and for the benefit of Muslims, and not by disposing of the very item purchased with this wealth, such as a house or a car.

The interpretation of Allah Almighty's saying: "And if you repent, then you shall have your principal sums" [Quran 2:279] prohibits usury, and indicates that a usurer, if he repents, is entitled only to his principal sum. Anything in excess he must return to its owner if possible, otherwise he should donate it as charity. And "you do not wrong, nor are you wronged" means you do not wrong by taking more than your due, nor are you wronged by having it diminished.