Is buying a house through a bank at a price higher than its original value, and in installments, with an initial down payment to the bank, considered permissible (halal) in Islamic jurisprudence, and what is the jurisprudential term for this transaction?

This transaction has two forms:

The first: The bank first buys the house, then sells it to a third party (the buyer) at a price that is purely free of usurious interest, even if paid in installments and at a higher price than the initial cost. This form is permissible and is classified as a Murabahah sale.

The second: The sale takes place directly between the buyer and the owner of the house, and the bank pays the price to the owner of the house, then records it against the buyer with an increase. This form is not permissible because it is a loan that yields a benefit.