Different people's usage of the terminology here differs. But I think the following way of talking is pretty standard:
- An interpretation is in the first instance an interpretion of a formal language $L$. Think of it as a map assigning truth-relevant values to the ingredients of $L$ -- for example, assigning objects to its names, extensions to its predicates, assigning a set of objects (a domain) for $L$'s quantifiers to run over, etc. (There is an assumed set of rules for determining the truth-values of sentences of $L$ in terms of such an assignment of truth-relevant values.) Derivatively, given a theory $T$ framed in language $L$, we can talk about an interpretation of that theory, meaning an interpretation of its language which assigns meanings-in-extension to the terms of the theory $T$.
- We are then particularly interesting in those intepretations of the language of the formal theory $T$ which make $T$ axioms and hence (assuming its logical apparatus is sound!) its theorems all true. Such interpretations are said to be models of the theory. Thus understood, a model is a value-assigning map between a theory and some objects, extensions, sets, whatever. Though often we carelessly speak about the assigned objects, extensions, sets, whatever, as being the model.
In this way of talking, not all interpretations of $T$ are models of $T$, only those interpretations which make $T$ come out true about the assigned subject matter. But models are interpretations, particular 'good' instances of interpretations (or in the derived usage, are the objects and sets etc. mapped to by the good interpretations).
When Wilder says a model (in the standard sense I have just defined) makes 'the statements of an axiomatic system to be true about a "concept"' that's either careless or wrong. If I interpret a formalized theory to be about beer cans and strings connected them, then the theory is about beer cans and strings. If I interpret a formalized theory to be about numbers and addition and multiplication, then that theory is about natural numbers, addition and multiplication. Beer cans and strings aren't concepts: neither (on most views) are natural numbers and operations of addition and multiplication.
But we could, in the second case, sum up our intepretation of the formal theory by saying it is about arithmetic, and that we have modelled our theory in arithmetic, and even that arithmetic is our model. But it doesn't seem at all helpful to say that the "concept" of arithmetic is the model -- its the numbers and operation themselves which are the ingredients of the model.
