This is my first encounter of simplex and simplicial complex, hence I am not very sure about the concepts.
In the definition of Simplicial Complex , I am not sure why is the second picture not a simplicial complex?
The intersection of the two triangles is a line, which is also a "face" isn't it?
As stochasticboy321 said, in the second example the intersection of the two simplices is a proper subset of the faces of each of them. A proper subset of a face is not a face!