Consider the following $\Delta$-complex homeomorphic to a torus, where the edges with the same orientation (i.e. the left and right sides of the square, and the upper and lower ones) are identified:
In some notes I'm reading it's stated that it's an example of a regular $\Delta$-complex, but I cannot see how it's possible.
According to the definition you've written, it doesn't seem to be regular - after identifying edges, there are only two distinct vertices. Therefore any $2$-cells can't possibly have three distinct vertices.