I'm afraid the use of the bra-ket formalism is not clearing up things, instead this use does what it usually does: hides some delicate mathematical issues.
Let's get it from the beginning. One of the key (not explicit) assumptions of the so-called 'orthodox' formulation of QM (due largely to Dirac and von Neumann through their 1930/1932 books) is that 'time is a parameter', not an observable. This folds nicely with classical dynamics in the Hamiltonian formulation: there, too the 't' is nothing but a parameter, not a classical observable (the phase space will not include it, instead we're interested in further parametrizing a classical state (point in the phase space) using 'time', time is then the intrinsic parameter of curves in phase space: evolution parameter of classical Hamiltonian states). So the nice Schrödinger building of a Hilbert space (ignore for now rigged Hilbert spaces or the notions of pure versus mixed states) will take 't' as a parameter (think actually of the Heisenberg picture, where the Hilbert space is <still>/<frozen>, while the observables such as momentum & position depend on time) in the following sense: At each moment in time, ## \psi (x)## is a normalized vector in a Hilbert space which is then (by the uniqueness theorem of von Neumann) safely to be taken as ##\mathcal{L}^2 \left(\mathbb{R},dx\right)## (assuming infinite motion).
So, in what sense do we interpret the LHS of the Schrödinger equation: ##\frac{\partial \psi (x,t)}{\partial t} ##? Well, in the very old way of treating partial derivatives of independent variables in an ordinary function of 2 (3,4, etc.) variables, i.e. using limits. The Schrödinger eqn. asserts that provided that
## \lim_{\delta t \rightarrow 0} \frac{1}{\delta t} \left[\psi (x, t+\delta t) - \psi (x, t)\right] ## exists, then it is equal to the vector in the codomain of the Hamiltonian operator at the same moment of time.
One can surely see the difference of 2 vectors taken before the limit as another vector whose norm then tends to 0 so that the limit is still finite (by postulation) and the result can even be normalized to unity. But one must understand that this limiting process is extraneous to the standard Hilbert space per se, i.e. this limit is not a weak, nor a strong one. Just as in classical dynamics, time acts as a parameter for curves of normalized unit vectors in the (complex, inf-dim., separable) Hilbert space. These curves are assumed to be at least of class C^1 in the parameter (differentiable of 1st order and the 1st derivative to be continuous).
As I wrote above, if you're willing to take the process of 1-time differentiation of a Hilbert space vector-valued function as a linear operator in the abstract sense, then, by virtue of the Schrödinger's equation, you can always compute the range of this 'linear operator': it's the vector you're getting by applying the Hamiltonian onto it.
So (as people call it nowadays here) FAPP, this linear operator is the Hamiltonian, a genuine operator, in the sense it's a function of the usual fundamental quantum observables: position, momentum, spin. You can even consider that this linear operator which takes a vector-valued parametric function to its derivative (also vector-valued) to have a hermitian adjoint in the regular sense. This adjoint is of course the adjoint of the Hamiltonian, which is of course the Hamiltonian, because as, again by postulating, the Hamiltonian is self-adjoint. The formal manipulation in the bra-ket formulation trivially follows.
