We have the following definitions:
A simplicial complex is a set of simplices so that any face of a simplex is a simplex and the intersection of two simplices is a face of both.
An abstract simplicial complex is a collection $K$ of subsets (simplices) (of a set $V$) so that every subset of a simplex is a simplex.
Is there any equivalence between the two? In particular, is there any way to obtain the intersection part of the first definition from the second definition? Or are these really just two different ideas?
Yes, the definitions are equivalent: see https://en.wikipedia.org/wiki/Abstract_simplicial_complex#Geometric_realization for a general explanation.
You specifically asked about the "intersection" part. Indeed, suppose an abstract complex $K$ contains two simplices $s_1, s_2$. The intersection $s_1\cap s_2$ is contained in both $s_1$ and $s_2$. By definition of an abstract complex, $s_1\cap s_2$ is as simplex in $K$, too. And it is a "face" (- a subset) of both $s_1$ and $s_2$.