Here my task is to build bridges between several different classical and contemporary uses of the word model, so I don’t have the luxury of complete control over the words in play but have to start from the customary senses in the various communities of interpretation. Of course I’m slyly working from a sign-relational backdrop, but I have to be sleight-handed about that and not hit people over the head with it.
You can probably guess that I’m using object to cover sign-relational objects, and theories are clearly syntacked together from complexes of sign-relational signs, so all we have left to pin down is where the various kinds of model sit at the table set with the labels of Object, Sign, Interpretant.
In its theoretical sense, a model of a theory is anything the theory is true of, anything that satisfies the theory. In that sense, a model is very like an object. It is whatever the theory is talking about. In the order of nature, indeed, models come before theories. But there is another order, the order of art, and one may construct artificial models out of almost any stuff, even the stuff of signs. So you see the kind of wiggle room we have to work with.
Things are easier outside of logic, in applied mathematics and the special sciences, where models are just things like analogues, icons, simulations, and similar representations of objects. But that makes them objects serving as signs of other objects, and so you may find some semiotic subtlety lurking there.