The notions are very different, to my best understanding.
As far as I could understand it, the pre-Cantorian infinity was primarily a notion of a length which is longer than any other. It was closer to the infinity we meet in real analysis, rather than the infinity we deal with in set theory. Despite the informal similarities that one can talk about "$f(\infty)$" and "$\alpha\in\sf Ord$", as if both these were actual objects of their universes.
Cantor noticed that one can consider a queue of infinite length, and that it makes sense to have an infinite queue, and then some. From there he defined the ordinals and the cardinals, and the rest is history.
On the other hand, from a modern point of view, the class of ordinals need not be absolute. One can consider end-extensions, which really add more ordinals to the universe, a typical example is taking an inaccessible $\kappa$, then $\bf V$ is an end-extension of $V_\kappa$.
In one considers theories like the Tarski-Grothendieck, or equivalently the theory $\sf ZFC+$"There is a proper class of inaccessible cardinals" then one can consider the least inaccessible as the universe of sets, and then one can always find a larger and larger universe with more and more ordinals. If one goes to even stronger cardinals (e.g. Woodin cardinals) then these properties of "extending higher and higher" can get stronger and stronger.
From another point of view, one can consider the multiverse approach which was proposed by Joel D. Hamkins (see J. D. Hamkins, The set-theoretic multiverse, Review of Symbolic Logic 5 (2012), 416–449.), which says among other things that every universe of set theory is really a countable model from the point of view of another universe (see also this blog post by Francois Dorais).
This is a much stronger assumption that the proper class of inaccessible cardinals. There we had a sequence of universes, each larger than the last, but none was countable in any of the extensions. In fact they all agreed on their common cardinalities and sets. In this case the universes become smaller and smaller as we go along.
So whereas the potential infinity of the early 19th century was somewhat of "the energy of an unstoppable object"; the class of ordinals is something vastly more frightening in size. But at the same time, the calculus notion of infinity is very coarse and hardly at all manageable. On the other hand, the class of ordinals is a concrete class (for a given universe, of course), which can be managed and manipulated internally. This is added by the above facts that the ordinals can be made a set of a larger universe; or even a countable set of a much larger universe.
At the very least, you should understand the statements of the completeness theorem, the incompleteness theorems, and the Lowenheim-Skolem theorem. This involves all of the basic definitions of formal logic and model theory; models, theories, satisfaction, elementarity, elementary substructures, Skolem functions, and more. Of course, as it is in any area of math, it's easy to trick yourself into thinking that you understand these things, when you really don't. You can get a more thorough understanding by reading the proofs, which make up the essentials of many logic texts.
I don't know the Chiswell-Hodges book, but glancing at the table of contents, it looks like it doesn't quite cover all of the prerequisites. (It may however be a very good book for what it does cover.) Enderton's book (on logic) is considered a classic, and it has approximately the right content, including the relevant definitions from model theory.
General model theory is a good thing to have an understanding of when going into set theory, but don't think it'll make the "Models of Set Theory" chapter of Jech a breeze; models of set theory are pretty counter-intuitive at first glance, and require a lot of thinking about on their own.
I agree with Arthur Fischer in the question you linked to; Jech is not a great book for a budding set theorist to learn the field from. It's more of a reference than a text. However, I think the above remarks apply to essentially any introduction to modern set theory.
Best Answer
First note that since $\mathsf{ZFC}$ is just a formal first-order theory, it doesn't pick out exactly what form a model's interpretation of the $\in$ symbol takes. So there are models of $\mathsf{ZFC}$ where the $\in$ relation is something other than the real membership relation. Actually, this point is crucial in the development of Boolean-valued models. In such circumstances, what $M$ thinks of as an ordinal will not be what $\mathbf{V}$ thinks of as an ordinal.
But I assume that you are thinking of models where $\in$ is interpreted as the real membership relation. But we do have the following facts.
In the second point above I mentioned transitive set models of $\mathsf{ZFC}$ (where $\in$ is interpreted as real membership). The existence of models of this kind is an even stronger assumption than the simple consistency of $\mathsf{ZFC}$, then we cannot show from $\mathsf{ZFC}$ (or even $\mathsf{ZFC}+\mathrm{Con}(\mathsf{ZFC})$) that such exist.
But transitive (possibly proper class) models of $\mathsf{ZFC}$ must agree with $\mathbf{V}$ about what ordinals are (if not "how many" there are):
This is due to the fact that there is a $\Delta_0$-formula $\phi(x)$ (i.e., a formula in which all quantifiers can be expressed in the form $(\exists x \in y)$ for $(\forall x \in y)$) which in $\mathsf{ZFC}$ defines the property of being an ordinal, and it is known that $\Delta_0$-formulas are absolute for transitive models.
A transitive model $\mathbf{M}$ of $\mathsf{ZFC}$ which contains all the ordinals (from $\mathbf{V}$) is called an inner model.