I've read on the website ncatlab.org that a pullback is defined as the subset of a cartesian product $A \times C$ of two sets $A \rightarrow B \leftarrow C$. I've also read about a pullback in the context of ' coordinate rings ' insomuch as a pullback of ' C - Algebras ' maps $u^2 + v^3 \in \mathbb{C}[W] \rightarrow W$ to $(x^{2}y)^{2} + (x - z)^{3} \in \mathbb{C}[V] \rightarrow V$.
I could not see which tuples make up the cartesian product though!
Osama Ghani has hit the nail on the head in the comments. Your definition that you've written in the post is of a categorical pullback, which is not appropriate in this setting. There is another notion of pullback: often times when one has a function, sheaf, or some other gadget defined on an object $Y$, given an appropriate map of objects $f:X\to Y$, we can produce a gadget on $X$ from the data of our gadget on $Y$ and the map $f:X\to Y$. This is frequently referred to as a pullback, and the method that defines it is called pulling back our gadget along $f$.
In our specific case, we're dealing with regular functions on a variety $Y$ and a map of varieties $f:X\to Y$. By the definition of a morphism of varieties, any regular function $\varphi$ on $Y$ defines a regular function $f^*\varphi$ on $X$ via the rule $(f^*\varphi)(x)=\varphi(f(x))$. This is called the pullback of $\varphi$ (by $f$), and it gives a morphism of coordinate algebras as defined at the top of the page.