In general, we don't distinguish carefully between the use of words like "theory", "model", etc., in the sciences. Most of the popular literature and discussions on the differences are overly-simplified bunk that groups like the AAAS have been trying to eradicate from textbooks along with the myths about The Scientific Method. There are uses of such words that are distinct in the literature across fields, but more important here is how the terms are used somewhat uniquely in physics specifically.
In physics as in many sciences, models differ in their use of parameters. The standard model of particle physics is just about the best theory in the whole of science, but it is also a model because of the parameters necessary for the theory to yield predictions. There is nothing that prevents models from being theories or the other way around, and both can be known to be quite wrong and still be called theories and models (as Box put it "all models are wrong, but some are useful").
Again, the distinction between model and theory here is not actually reflected in any of the literature nor is it very accurate. There are plenty of theories that are highly speculative in cosmology, and some of them are models. In foundation physics, cosmology, and a slew of other fields in physics that deal with quantum theory, relativistic physics, ,and field theories, one often finds the word "theory" used to mean "Lagrangian". Of course, the Hamiltonian and Lagrangian formulations are sort of meta-frameworks in that they are primarily differing ways to apply variational principles along with the action principle to physical theories like quantum mechanics or classical gauge theories or non-relativistic QFT or the product space of representations underlying the standard model and so on.
In this way special relativity is a more general framework than general relativity (despite the latter being a "generalization" of the former) specifically because it is not a theory of anything. It is a framework within which one can formulate both quantum and classical field theories in terms primarily of the Lagrangian framework and point-particle mechanics like QM in terms e.g., of the Hamiltonian (classical electrodynamics is automatically relativistic as the necessary symmetries and invariances are built in to the theory and indeed are largely a consequence of Maxwell's equations). General relativity, on the other hand, is a theory of gravity. It is a much better theory of gravity than Newton's gravitational force. But unlike quantum mechanics, it comes with built in failures that ensure its own demise irrespective of how it will breakdown on microscopic scales or how it cannot be made consistent with quantum theories.
There isn't just one big bang cosmology, unless you are conflating the classical version with inflationary cosmologies (which are generally accepted and are certainly scientific theories). The big bang theory isn't actually a theory but a consequence of one combined with a name Hoyle used to ridicule the idea.
There are several multiverse cosmologies. One is eminently testable as it is an interpretation of quantum mechanics. Others are almost certainly correct according to all of our best evidence but are misleading in how they are named. None are really models, of course, as they are classes of theories (some of which include models or are based on models. They different theories tend to be derived from differing assumptions in the mathematical structure and interpreations of empirical observations/scientific data that yield different theories. Different models would be obtained by plugging in different parameters to the same theory or something like one.
String theory started out as a failed attempt to work through problems in Hadronic physics when it appeared that QFT, after its initial successes with QED, was completely inadequate and had to be abandoned. This was at a time when the bootstrap programme and s-matrix theories were still big candidates over and against field theorists. Eventually, with the acceptance of the Quark model and the failure of string theory to even adequately deal with confinement at all, it was temporarily abandoned and QFT became and remains the only working theory explaining elementary interactions via the standard model. String theory is now a name given to a slew of mostly vague ideas with some neat, non-rigorous mathematics which are hoped to eventually move from intuitions to some kind of solid, reasonably well formulated mathematics.
Currently, the math is not better than good, it is about as bad as it is possible to be. It is mostly all problems. String theory is horribly overdetermined as one of the many possible attempts to get off the ground with something like an effective theory failed when it was proved that any attempt at compatification had to face the fact that there are necessarily infinitely many possibile "solution" spaces for any would-be formulation. Nothing in string theory suggests the possibility of removing the divergences which plague the standard model and QFT currently, but rather so far we have more difficult mathematical problems instead of the solution to something that would either make renormalizability coherent and consistent or remove the need for it altogether (either of which is necessary for quantum gravity, as it is non-renormalizable). String theory is essentially ~50 years of failed mathematical attempts which yielded some tremendous results in pure mathematics, but not mathematical physics or theoretical physics.
Quantum mechanics cannot account for particle annihilation or creation which are necessary for any kind of attempt at explaining any fundamental interactions, including quantum electrodynamics or even just photons, and the field theories we have currently are called effective field theories precisely because they work find up to some scale at which point we insert our cut-off parameter and sweep the infinities predicted by the theories under the rug. This is just find. It is more than fine. It is tremendously accurate. But it is not quantum mechanics, which treats time as a parameter and position as an operator. QM can be made consistent with special relativity and this is what most quantum field theories are: QM with fields in 4-space where position is demoted from an operator acting on the system's state.
GR isn't as successful as special relativity. It is an excellent theory of gravitational force. But it breaks down in numerous ways even in its own domain and allows for ridiculous solutions that cannot be ruled out within the theory (or any other theory) even though they may involve blatant causality violations. And we still don't have a very consistent way of interpreting (or even really necessarily defining) most of the singularities in the theory.
It was never a model, but a theory. It is still a theory. It was always a theory of gravity. It remains a theory of gravity. It is a theory that has been tested and as a tested theory of gravity it is our best theory of gravity and a great one.
