"Proper" definition of a proper map?

A continuous map $f:X \to Y$ between topological spaces is proper if it is universally closed, i.e., for all $T \to Y$ the induces map \begin{align} X \times_Y T \to T \end{align} is closed. This definition is equivalent to the one you mentioned when $X,Y$ are locally compact Hausdorff. Here $X \times_Y T$ denotes the fiber product in the category of topological spaces (it is just the subspace of $X \times T$ where the maps to $Y$ agree).

A morphism $f: X \to Y$ between schemes is called proper if it is separated, finite type and universally closed. That means that for all morphisms $T \to Y$ the map \begin{align} X \times_Y T \to T \end{align} is closed. This time, $X \times_Y T$ denotes the fiber product in the category of schemes (or locally ringed spaces).

The connection is as follows: Let $a:X \to \operatorname{Spec} \mathbb{C}$ be a smooth proper variety ($a$ is a proper morphism), then the complex manifold $X^{an}$ is compact. More generally, if $f:X \to Y$ is a proper morphism between smooth proper complex varieties, then the analytification of $f$ is a proper morphism of topological spaces (I don't think the smoothness assumption is at all necessary)

In Stacks project, maps satisfying (1) is called quasi-proper and (2) is called proper. (2) is equivalent to being universally closed (proved in Stacks project).

However, it seems natural (at least to me) to define a proper map to be universally closed and separated. This definition is equivalent to (3) (being relatively Hausdorff is equivalent to being separated) and is an analogue of proper morphisms of schemes. Another reason to adopt this definition is that extremally disconnected spaces are exactly SProp-projective objects in $\mathrm{Top}$, where $SProp$ is the class of all surjective proper maps. (See Stone spaces, Johnstone).