A diagram which is not the torus

At page 17 of Munkres' Elements of Algebraic topology it says, referring to fig. 3.6, that "the diagram does not determine [the torus]. It does more than paste opposite edges together":

Is it so? I tried to glue them together, but end up with a normal torus.

Let's count vertices, edges and faces.

The vertex-set is $\{a,b,c,d,e,f\}$. So $V=6$.

The edge-set is $\{ab,ac,ad,ae,af,bc,bd,be,bf,cd,ce,cf,de,df,ef\}$. So, $E=15$.

The face-set is $\{abd,abe,acd,acf,ade,adf,bce,bcf,bde,bef,cdf,cef,\}$. So $F=12$.

This gives the Euler characteristic as $\chi=V-E+F=6-15+12=3$. It follows that this space can not be the torus as the torus has Euler characteristic equal to $0$.

The reason this doesn't work as a cellular model for the torus is that there are edges which are identified but which we would not want to identify in the usual model of the torus. For instance, the two appearances of the edge $be$ in the model mean we would identify edges which we would ordinarily consider as lying on the surface of the torus as two intervals which compose a meridional circle.