The material I'm reading from is located at:
https://pdfs.semanticscholar.org/e508/b945c5c95fb4ac5810a180536be3b6292743.pdf
I'm confused how a model can contain elements.... I thought that a model is nothing more than an interpretation that makes everything in a certain set true. In other words, given an interpretation $M$ we say that $M$ models $\Delta$ iff $M\models \phi$ for all $\phi \in \Delta$.
Still, however, the article (responding to Hilary Putnam) speaks of models containing elements
Now, because $ZFC \vdash \exists x "x \text{ is uncountable}" $, there must be some m^ $\in M$ such that $$M\models "\text{m^ is uncountable}"$$ However, since $M$ itself is only countable there are only countably many $m\in M$ such that $M\in m\in \text{m ^}$
So maybe there are just different uses of the word "model"?
An interpretation has one essential component you've forgotten: a "domain of discourse", which is the "universe" the interpretation is working in. This is what allows the interpretation to make sense of sentences that involve quantifiers - how else can you interpret a statement $\forall xP(x)$, when not all objects in the universe are required to be named?
In set theory in particular (but in other fields as well) we often fail to distinguish between a model and its domain of discourse. For example, in set theory, we usually take it as implied that a model interprets in the "natural" way - that is, for $a$ and $b$ in the domain of discourse of $M$, $M \models a \in b$ if and only if $a$ is really a member of $b$. So, for a model of set theory, you really don't need anything beyond the domain of discourse.