How do I know the input current of a LED driver?
In the first situation the system draws 275 mA, in the second
  situation it draws 10 mA.


Second situation current will be 285 mA.  

The LED driver manufacturer can only tell how much the driver will consume itself (for the driver to work properly) and how much is the LED current it can support for a given set of condition.  

The second part in the image is what the IC can support. You have to consider 330mA plus 12 mA as the output for your DC DC regulator.