240v ac to 12v dc transformer voltage drop under load
This isn't a transformer, it is a switching power supply.

The "12V 9A" pump is most likely powered by a DC motor, if 9A is the nominal current then it will draw a lot more than 9A when starting. Perhaps 5x more at least.

A 12V battery won't have any problem with this. A 12.5A switching supply will cut off as its overcurrent protection gets triggered. The motor stopping and starting in your video corresponds to the power supply's protection tripping and then resetting after a time delay.

You either need a much more powerful supply (like a 450W PC power supply with most of the power available on the 12V output), or a mains powered air pump, or keep using the 12V battery.