Is extending a theory by a totally random/arbitrary predicate a conservative extension?

Suppose that I take ZFC and extend it by my own predicates like Fish(x) and Flies(x), and augment it with some new axioms, e.g. $\forall x(\text{Fish}(x)\to\text{Flies}(x))$.

I am having trouble understanding various exact statements of conservative extensions. Although I have a vague idea of numerous explanations of what's going on, they all get too technical for me, so I thought that maybe its best to instead ask about an example, like the one above.

Some places make it sound like an extension is conservative if it expands the language and/or axioms/rules of inference of the old theory, but all theorems of the new system which are written only in the symbols of the old language are already theorems of the old system. Technically since adding the axiom $\forall x(\text{Fish}(x)\to\text{Flies}(x))$ doesn't allow you to prove any new statements in the language of ZFC, is it a conservative extension? It certainly doesn't feel that way, but is it?

Idk what to tag this question but I've been trying to learn model theory on my own just in order to answer it, and have nearly given up.

As an update, after lot of reading about posts like this, I want to say that the answer seems (in heavy italics) to be yes, but I really have no intuition as for why this should be the case. The only thing I can find explicitly to the contrary is the wiki article on extensions by definitions, which kind of implicitly says that new statements need or have some translation or another back into the old language. But I still don't really know what the right answer to this is, about whether or not the extension in the opening paragraph, for example, would be considered as conservative.

It certainly doesn't feel that way, but is it?

This is absolutely a conservative extension. In fact, it's a conservative extension in a very strong sense: every model $M$ of $\mathsf{ZFC}$ has an expansion to a model of your theory.

I suspect that your concern comes from the sense that the behavior of the symbols you introduce is unconstrained by $\mathsf{ZFC}$ alone; model-theoretically, every model $M$ of $\mathsf{ZFC}$ has many distinct expansions to models of your theory. This doesn't contradict conservativity, however; it just means that your theory is not an expansion by definitions of $\mathsf{ZFC}$.