In the opposite direction any diagram $(\sum_ g, \alpha_1, ..., \alpha_g, \beta_1, ..., \beta_g)$ where the $\alpha$ and $\beta$ curves satisfy the first two conditions in Definition 2.3 determine uniquely a Heegaard decomposition and therefore a 3-manifold.
Why is statement correct? This statement is in "An introduction to Heegaard Fleor homology" article, page 4, written with ozvath and szabo.
Suppose that $\sum_g$ be a torus. if attach a $D^2$ along $\alpha$, remaineded a $D^3$, this remainded space uniquely fill by a $D^3$ and for $\beta$ we can doing manner again. Similar we can do it for$\sum_g$ be a surface by genus g.