How to identify the correct power supply for a device
You have to find a power supply that gives 12 V, also make sure the pins that connect to the plug (not in the socket but in the device) has the + and - correctly (some barrel plugs have the + as the inside pin, and the - on the outside, or vice versa). 
Looking at the picture your supply expects the + on the inside pin.

The power, should be AT least the one that is mentioned on the Chinese supply, which is 1A. It does not matter if it can handle more current/power, but it is a problem if it can deliver less. The 0.5A is the output of the wall outlet (which seems really high, but does not matter in this case). It would mean that your 100-240V socket takes 0.5A which means 100 V * 0.5 A = 50 W which is much more than the output of 12 V * 1 A = 12 W... so either the 0.5 A is slightly a too high value, or the supply is really inefficient (could be both).

The lamp will require current, if the supply cannot deliver it, it either overheats, stops working or some safety measurement will apply. In case of a Chinese power supply, it might overheat, which can result in fire.

So to summarize, find a power supply that:


Has 12 V
Has at least 1 A
Has the + of the barrel on the inside pin