My own preference is for a notion of possible world which I would guess is pretty much the kind of thing a mathematical physicist would go for.
First let me observe that I don't see any way we can get a handle on the real world other than through various kinds of models.
So I'm just going to identify "possible world" with "model of a world", and then a bit of discussion of how to build mathematical models.
I'm going to split the building of a model into two phases, which I will call metaphysics and physics for reasons which I hope will be fairly clear.
In the first phase we decide what kind of thing is a possible world, and then in the second we try and pin down the actual world.
There is no single correct answer in either phase, but the ways in which answers can be incorrect differ in the two phases.
In the first phase we are essentially doing mathematics, leading to a definition of the concept possible world.
If we first adopt and later reject a definition, we do not consider ourselves to have formulated a conjecture which we have later discovered to be false.
Instead we proposed a usage, which we later concluded was not such a good idea, just as mathematicians might debate which among alternative definitions of a concept is likely to be most fruitful without admitting that a definition can be true or false.
The metaphysics splits again into two parts, which I will call abstract ontology and concrete ontology.
The former is mathematics, the latter metaphysics proper.
For our abstract ontology we might take the cumulative hierarchy of sets, or some other adequate ontological basis for mathematics.
None of these entities are in themselves considered constituents of a material world, but our metaphysic will identify some of them which are to be interpreted as models of possible worlds.
To make this a little more definite lets consider a classical abstract ontology (just that of standard well-founded set theory), and a metaphysic which I will call Newtonian.
The standard set theory permits us to construct the real numbers, and vector spaces.
A point mass newtonian model state is a set of point material objects each of which has three spatial coordinate which are real numbers, and three velocity components, also real numbers.
The natural laws of a possible point mass newtonian world are modelled by a function which maps a model state to a set of triples of differential coordinates, one for each point mass in the state.
These indicate the rate of change of velocity of the point mass.