From the point of view of the condensed matter physics, the distinction between strong coupling and strongly correlated seems rather evident, and largely agrees with the point of view expressed in the question. First of all, this is because in the condensed matter physics the coupling strength is not fixed by the world constants but material-dependent, which allows exploring regimes with different coupling. Moreover, strongly correlated systems are often treated as a subfield of condensed matter, rather distinct from the rest. Let me give a few examples.
The polaron problem
The problem is about the kinetic and thermodynamic properties of an electron dressed by a phonon cloud, rather similar in spirit to many QFT problems
$$ \hat{H} = \frac{\mathbf{p}^2}{2m} + \sum_\mathbf{q}\hbar\omega_\mathbf{q}a_\mathbf{q}^\dagger a_\mathbf{q} + \sum_\mathbf{q}g_\mathbf{q}\left(a_\mathbf{q}e^{i\mathbf{qr}} + a_\mathbf{q}^\dagger e^{-i\mathbf{qr}}\right)$$
It has been extensively studied both in weak-coupling and strong-coupling regimes, either of which is realizable. We literally talking here about the magnitude of the coupling constant.
Ising model
Ising model is the main toy model in the field of strongly-correlated systems:
$$
H = -J\sum_{\langle i, j\rangle}S_i S_j - H\sum_i S_i
$$
Speaking of strong correlations one means that spatial and temporal correlations between parts of the system are decaying slower than exponentially (typically as power laws). Big part of this field is about studying phase transitions: the exchange coupling $J$ in the Ising model can be small, yet below the transition temperature the model will exhibit long-range correlations (in two or more dimensions). One must note that the transition temperature is dependent on the size of coupling, so it is hard to call it strong or weak - it is not clear what it must be compared with.
Kondo model
The examples above actually demonstrate that terms strong coupling and strong correlations are often applied in different contexts: the former often deals with a small system coupled to a bath (e.g., electrons and phonons) where coupling strength mediates the interaction between the two, the latter deals with collections of many identical interacting entities, where coupling strength is hard to identify.
An example at the intersection of these two is the Kondo problem of an impurity spin coupling via exchange interaction to the electron sea:
$$
\hat{H} = \sum_{\mathbf{k}\sigma}\varepsilon_{\mathbf{k}\sigma}c_{\mathbf{k}\sigma}^\dagger c_{\mathbf{k}\sigma} - \mu_B\mathbf{H}\cdot\mathbf{S} - \sum_{\mathbf{k}_1\sigma_1,\mathbf{k}_1\sigma_2}\mathbf{S}\cdot\mathbf{s}_{\sigma_1\sigma_2}c_{\mathbf{k}_1\sigma_1}^\dagger c_{\mathbf{k}_2\sigma_2}
$$
It can be characterized in terms of a coupling strength, yet it exhibits many features of the strongly correlated systems. Again, the transition to the strongly correlated regime is determined by the Kondo temperature $T_K$, which depends on the coupling strength $J$.
