$$\frac{d\mathcal{H}}{dt}= \dot{q_i}\frac{\partial\mathcal{H}}{\partial q_i}+\dot{p_i}\frac{\partial\mathcal{H}}{\partial p_i}+\frac{\partial\mathcal{H}}{\partial t}$$
And physically this is related to how the Hamiltonian experienced, $\mathcal{H}(t)=\mathcal{H}(q_i(t),p_i(t),t),$ changes as you move around in phase space, compared to how the Hamiltonian field $$(q_i(t),p_i(t))\mapsto \mathcal{H}(q_i(t),p_i(t),t)$$ changes in time.
And as you pointed out $\frac{d\mathcal{H}}{dt}= \frac{\partial\mathcal{H}}{\partial t}$ means $\dot{q_i}\frac{\partial\mathcal{H}}{\partial q_i}+\dot{p_i}\frac{\partial\mathcal{H}}{\partial p_i}=0.$
Which means the flow in phase space is orthogonal to the gradient of the Hamiltonian. That the Hamiltonian experienced isn't changed by the flow in phase space, but solely by the change of the Hamiltonian field in time.
This is not the same as Hamilton's equations. Because it does not, for instance, imply Hamilton's equations. It is a consequence of Hamilton's equations.