Strong and classical are not the same. Classical should be clear: a point-wise solution of the equation, and therefore a function $u$ of class (at least) $C^2$ that satisfies (*) at each point of $\Omega$.
A strong solution is something different, at least in principle: it is usually a twice weakly differentiable function $u$ that satisfies (*) almost everywhere. This is the definition in Gilbarg, Trudinger: Elliptic partial differential equations of second order, $2^{\mathrm{nd}}$ edition.
Edit: in particular, the concept of strong solutions is confined to variational equations. However, it seems that other Authors write strong to mean classical.