which came with a power supply (see photo). The power supply is rated at 11V at 0.5A, 5.5W max, max 12V.
The cable was too short so I bought a new power supply from amazon (see photo) which says it's 12V, 0.5A.
Recently the lamp has stopped working and I'm wondering whether it was anything to do with the new power supply. Could it be? I thought it would be OK.
Sorry to tell you – you've destroyed the lamp. The original power supply is a constant current supply specifically for driving LEDs.
The replacement is a typical constant voltage power supply. With a constant voltage supply, the LEDs will draw as much current as the supply will deliver and burn out.