Ice is not water, because water is Fluid, Water fills the shape of whatever container you put it in, Water quenches your thirst. Since Ice does not do any of this we can say it is not water? Ofcourse not, it is merely Water in a separate state, and because it is in this separate state it gained properties that Water does not have and does not have properties that Water does have. And yet Ice is still Water.
Amaterasu is to Fire what Ice is to Water, merely Fire on a separate State. All Fire needs is Fuel, in a regular instance Fire uses Oxygen as Fuel, in the case of Amaterasu the Fuel it uses is the Chakra of the user.