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Cut the second circle factor at $p,q$. This divides the torus into two annuli whose boundary circles are exactly the two collapsed circles. Collapsing each annulus's two boundaries separately produces a two-sphere; the two spheres then share their two poles. In particular, the two collapsed circles give two distinct vertices, not a single common collapsed point.

For an explicit cellular homology calculation, each suspended circle contributes one edge from the first vertex to the second and one two-cell attached with zero cellular boundary. Thus $C_0\cong {\mathbb{Z}}^2$, $C_1\cong {\mathbb{Z}}^2$, $C_2\cong {\mathbb{Z}}^2$, with no higher cells. Both edges have boundary $v_q-v_p$, so ${\partial }_1$ has rank one and kernel generated by the difference of the edges. Each two-cell's attaching loop traverses its edge in opposite directions, so ${\partial }_2=0$. Consequently the homology of a torus with two parallel circles collapsed isCollapsing either connecting edge as a maximal tree also exhibits the homotopy type $S^1\vee S^2\vee S^2$.