Which "version" is this definition of Simplex and Simplicial Complex

Question

I read this definition of simplex on an arxiv paper: https://arxiv.org/pdf/1709.10198.pdf.

I am curious which "version" of definition of simplex are they using? It seems different from the usual definition of abstract simplicial simplex: https://en.wikipedia.org/wiki/Abstract_simplicial_complex.

Thanks.