If the bank authorizes me to buy the car myself, is this still considered a Murabaha transaction?

It can be a Murabaha transaction even if the bank appoints you as their agent to execute part of the purchase.

Steps

1. However, agency means you are acting on behalf of the bank at this stage, not that the bank simply gave you money and then labeled the transaction as Murabaha.

2. The important thing is that the steps of agency, purchasing the asset, and then selling it to you are separate and clearly documented.

3. If you're unsure whether you're signing as an agent or a buyer with each signature, ask before signing.

Source

• AAOIFI — Sharia Standard 8: Murabahah