In Shafarevich's Algebraic Geometry 1, he defines a proper map between quasiprojective varieties, namely $f: X \to Y$ as a map $f$ that can be factored into the composition of $\iota: X \hookrightarrow\mathbb{P}^n \times Y$, a closed embedding and $\pi: \mathbb{P}^n \times Y \to Y$, a projection.
My definition of a proper map in the sense of topological spaces is that the preimage of compact sets is compact, or an equivalent definition in locally compact and Hausdorff spaces, the map is closed with compact fibers.
My question is how are these definitions related? How could I prove that these are equivalent definitions in the case where $f: X \to Y$ is a regular map of quasiprojective varieties?
In the text, he goes on to say that the preimage of a point is a projective variety, (in particular it will be compact), so since $\iota$ and $\pi$ are closed maps then we have that $f$ is a proper map (in Shafarevich sense) implies that $f$ is a closed map with compact fibers. Is it possible to show the converse?
Relevant post, but a different question: Is a proper morphism between quasiprojective varieties projective?
Thank you for your time!