Clairaut's theorem states:
"Suppose $f$ is defined on a disk $D$ that contains the point $(a,b)$. If the functions..."
My question is just about the first part of this. What does it mean for a function to be defined on a disk?
Does it just mean that that disc (which is a always a subset and always(?) a subspace of $\mathbb{R}^2 $) is the domain of the function in question? In that case, doesn't defining $f$ only make sense for a function of two variables and nothing else?
No, it means that $f$ must be defined at least on a disk (a filled in circle) centred at $(a, b)$. It may be defined elsewhere too. It doesn't really matter for the purposes of determining derivatives anyway; only points around $(a, b)$ will determine its various derivatives.