In my geometry notes the writer states that the following two, are not triangulations for the torus:
On the contrary this is a good triangulation:
I tried to wrap a piece of paper in order the check the issues with the first two examples, but I'm not a good artist! As I could read here in a similar question, the first one can't be a triangulation since the $AD$ segment becomes a circle. By the way in the second example this doesn't happen since every edge is divided into two pieces. What's the problem with the second one?
A triangulation has the property that two simplices are only allowed to intersect in another simplex (For the two dimensional case this is a point or edge). In the first picture the two triangles intersect in a union of three edges, which is not a simplex. I think you can see what goes wrong in the second picture now.