From Commutative Algebra - Constructive Methods by Lombardi and Quitte:
Definition 2.9. A property $\mathsf P$ concerning commutative rings and modules is called a finite character property if it is preserved by localization and if, when it holds for $S^{-1}A$ then it also holds for $A[\frac 1s]$ for some $s\in S$.
The definition I know goes as follows:
Definition. A formula $\varphi$ with one free set variable has the finite character property if $\varphi (\emptyset)$ holds, and for every set $A$, $\varphi (A)\iff \varphi (F)$ for every finite $F\subset A$.
What is the relationship between these notions? Moreover, what is "the correct" formalization of 'property' in the former definition? (I don't know logic.)
These notions are not related. They just happen to have the same name.
To answer your second question:
The typical way to formalize the notion of a property $P$ is by a formula $\varphi_P(x)$ of your formal system expressing this property (so that an object $A$ satisfies property $P$ if and only if $\varphi_P(A)$ is true). So, for example, if we're formalizing a property of rings in ZFC set theory, $\varphi(x)$ will be a formula of first-order logic in the language $\{\in\}$ of sets, which accepts as input a ring, encoded as a set in a standard way.
Of course, if you're just trying to do commutative algebra, you don't need to get too caught up in verifying that all your favorite properties are expressible in the language of set theory (though it's good to think through some examples like this when you're learning set theory!). One of the things we want from a good foundation for mathematics is that it's expressive enough to capture all the properties that we talk about in the informal language we actually use when talking about math.