quantum mechanics - Why can't $ihbarfrac{partial}{partial t}$ be considered the Hamiltonian operator?

In the time-dependent Schrodinger equation, $H\Psi = i\hbar\frac{\partial}{\partial t}\Psi,$ the Hamiltonian operator is given by

$$\displaystyle H = -\frac{\hbar^2}{2m}\nabla^2+V.$$

Why can't we consider $\displaystyle i\hbar\frac{\partial}{\partial t}$ as an operator for the Hamiltonian as well? My answer (which I am not sure about) is the following:

$\displaystyle H\Psi = i\hbar\frac{\partial}{\partial t}\Psi$ is not an equation for defining $H$. This situation is similar to $\displaystyle F=ma$. Newton's second law is not an equation for defining $F$; $F$ must be provided independently.

Is my reasoning (and the analogy) correct, or is the answer deeper than that?