Let $[x_0,...x_n]$ is an n-simplex of $\mathbb{R}^{n+1}$. Then I have a definition for the j$^{th}$ face as the following:
$\{{\sum_{i=0}^{n}{t_ix_i}}\in{[x_0,...,x_n]}|t_j=0\}$
This makes sense in the example of the triangle simplex of $\mathbb{R}^2$ but my confusion lies in the following example:
Suppose the simplex we are discussing is non-degenerate simplex that looks like a square on the plane. If we set a $t_j$ to 0, would the jth face not be then a right angle triangle with corners $\{x_i|i\neq{j}\}$ instead of what you'd expect it do be (one of the sides of the square)?
I think this is the result of some confusion between an $n$-simplex and an n-cell.
An $n$-simplex is defined to be the convex hull of $n$ linearly independent vectors in $\mathbb{R}^{n+1}$. These will always be triangle-things (line, triangle, tetrahedron, ...). An $n$-cell is (something homeomorphic to) the $n$-dimensional disk.
You may object that these are the same thing in the sense that they are homeomorphic. However, we occasionally want to use the additional structure of simplices, most notably in simplicial homology (and in your case, the face map).