Based upon your description, my guess would be that you connect either red wire to the black wire from the lamp, and the other red wire to the white wire of the lamp.
Then connect the black wire from the transformer to the black 120V power cable, and the white wire from the transformer to the white 120V wire.
But, the instructions make me see how you got confused.
Can you post a pic of the markings on the transformer, or describe them if not?
Then connect the black wire from the transformer to the black 120V power cable, and the white wire from the transformer to the white 120V wire.
But, the instructions make me see how you got confused.
It sounds like the wires to the lamp should have been factory connected..? I have a hard time believing that the white wire from the transformer is 12V, though who knows.
Can you post a pic of the markings on the transformer, or describe them if not?