I don't see the difference between "domain of definition" and "domain of definition for the solution".
The fact that the denominator is factored actually gives us a clue.
There is a theorem that states: If $\frac{dy}{dt} = f(t, y)$, $y(t_0) = y_0$, where both $f$ and $\frac{\partial f}{\partial y}$ are continuous in some open rectangle $(a, b) \times (c, d)$ containing the point $(t_0, y_0)$, then there exists a unique solution to the IVP for some interval of $t$-values, $(t - \epsilon, t + \epsilon) \subseteq (a, b)$.
Now this implies that we can only expect a solution to your ODE somewhere within the infinite strip defined by $-1 < y < 2$, $t \in \mathbb{R}$. If the solution curve leaves this strip, then all bets are off. In practice, one finds the appropriate restrictions on $t$ based on the solution of the IVP (if one can be found). Fortunately, your ODE was separable, and so just plugging in $y = -1$ and $y = 2$ will give the appropriate $t$-bounds. (By the way, you can determine $C_1$, since an initial value is given.) It helps to graph your solution... Graph $t = y^3/3 - y^2/2 - 2y$ and think "inverse functions".
Hope this helps!