Suppose I define a function as $f(x,y)=2(x+y)$. Compare that definition to $f:\mathbb{R}^{2}\rightarrow\mathbb{R}, f(x,y)=2(x+y)$, which also gives the (co-)domain.
Is there any standard way to refer to these definitions? Is the second one a "complete" or "full" definition?
If one wants to be very precise then the definition of a function always contains the information of the domain and the codomain/range/image of the corresponding map.
Often this is clear from the context, so it is often suppressed for a more convenient notation. Nevertheless, technically one needs to state everything - so actually there is no real complete or half definition, just the definition which contains all the information.
In your example we could assume the following scenario: $$ f:\{(x,y):x^2+y^2=1\}\subset \mathbb{R}^2\to \mathbb{R},f(x,y)=2(x+y) $$ which is clearly different from $$ f: \mathbb{R}^2\to \mathbb{R},f(x,y)=2(x+y) $$ and this is again different from $$ f: \{(x,y):x^2+y^2=1\}\subset\mathbb{R}^2\to \operatorname{im}(f),f(x,y)=2(x+y) $$ this becomes obvious when you ask for properties like injectivity, surjectivity etc.