When I think of a "simplicial complex", I think of the geometric realization of a simplicial set (a simplicial object in the category of sets). I'll refer to this as "the first definition".

However, there is another definition of "simplicial complex", e.g. the one on wikipedia: it's a collection $K$ of simplices such that any face of any simplex in $K$ is also in $K$, and the intersection of two simplices of $K$ is a face of both of the two simplices. There is also the notion of "abstract simplicial complex", which is a collection of subsets of $\{ 1, \dots, n \}$ which is closed under the operation of taking subsets. These kinds of simplicial complexes also have corresponding geometric realizations as topological spaces. I'll refer to both of these definitions as "the second definition".

The second definition looks reasonable at first sight, but then you quickly run into some horrible things, like the fact that triangulating even something simple like a torus requires some ridiculous number of simplices (more than 20?). On the other hand, you can triangulate the torus much more reasonably using the first definition (or alternatively using the definition of "Delta complex" from Hatcher's algebraic topology book, but this is not too far from the first definition anyway).

I believe you can move back and forth between the two definitions without much trouble. (I think you can go from the first to the second by doing some barycentric subdivisions, and going from the second to the first is trivial.)

Due to the fact that the second definition is the one that's listed on wikipedia, I get the impression that people still use this definition. My questions are:

Are people still using the second definition? If so, in which contexts, and why?

What are the advantages of the second definition?