It is an abuse of notation: it mixes the <PERSON> notation (i.e. the usual wave function representation) with the abstract Dirac notation (the bra-ket notation). In your example, the Hamiltonian operator is expressed in the <PERSON> notation (i.e. with position operator that is "diagonal" and the momentum operator represented by the gradient), while the "state" is expressed in the <PERSON> notation (it is an abstract ket).
Long story short:
State at a particular time $t$: it is an abstract container $| w, t \rangle$ for some information $w$ at time $t$ ($w$ is a collection of values and/or procedures that determines the state of your object.. some sort of recipe to "prepare" it).
Wave function relative to the info $w$ at time $t$: think about is as a complex function $\Psi_w(x,y,z,t) = \langle {\bf x}| w, t \rangle $. Usually $w$ is dropped (in fact you are not even able to write it down most of the times, and it is a formal label for the state) and the common notations $\Psi({\bf x},t) = \langle {\bf x}| \Psi, t \rangle$ or $\Psi({\bf x},t) = \langle {\bf x}| \Psi(t) \rangle$ are used. You can think of $\Psi$ (or whatever name you want to use as something that completely labels the state and that is directly used in place of $w$). 976
Note: $| {\bf x}\rangle$ is the abstract eigenstate of the position operator relative to the eigenvalue ${\bf x}$ (in the case of this "position ket" the information $w$ is simply $w={\bf x}$, namely "the particle is in ${\bf x}$"). In other simple cases the label $w$ may be a set of "quantum numbers" or eigenvalues.
Completeness: since (formally) you have that $\sum_{x,y,z}| {\bf x}\rangle \langle {\bf x}|$ is the identity operator,
$$ |\Psi,t\rangle = \sum_{x,y,z} |{\bf x}\rangle \langle {\bf x}| \Psi(t) \rangle
= \sum_{x,y,z} \Psi({\bf x},t) |{\bf x}\rangle \, , $$
which means that the ket $|\Psi,t\rangle $ and the associated wave function carry the same info (are both complete and equivalent representations of the state.. the ket one by definition). Moreover,
$$ \partial_t |\Psi,t\rangle = \sum_{x,y,z} (\partial_t \Psi({\bf x},t) ) |{\bf x}\rangle $$
Bad notation: on the other hand, $\nabla |\Psi,t\rangle$ is just abuse of notation. To make sense of this, the gradient must be understood as an "abstract operator", acting on kets, that has the effect of differentiating wave functions once the expression is projected onto the position basis: you can try to use the completeness to obtain the decomposition of the abstract momentum operator (EDIT: see the nice answer of Qmechanic for this point!).
Time VS space: the derivative in time, although formal, is not an abuse of notation if applied to kets. In fact, time is not the eigenvalue of something, position yes. QM treats time and space in fundamentally different ways. Time is just a parameter in QM, so you can think of $|\Psi,t\rangle$ as a curve in the abstract space of states, and this curve has a "tangent" that is formally indicated as $\partial_t |\Psi,t\rangle$. 669