Exactly right. Check out the
definition of a triangulation - it's a homeomorphism between a simplicial complex and whatever space you're triangulating. A simplicial complex is just a glued-together collection of simplices, which are just triangles / tetrahedrons / higher-dimesional versions of the same. Now, on the surface of a torus, two triangles can share all three vertices (as demonstrated above). But simplices live in R
^{n}, where such a thing is impossible. Therefore, what you've mapped into that torus isn't really a simplicial complex, so it's not really a triangulation.