A typical classification result for a class $C$ of objects looks like that:

Theorem. Each object of $C$ is isomorphic to one object of the following list: [insert list here].

Examples are the classification of finite simple groups and the classification theorem of closed surfaces.

Now I noticed sometimes people also call theorems of the following type "classification results". Here, $I$ is a sufficiently strong invariant of the class $C$ of objects.

Theorem. Let $X$ and $Y$ be two objects of $C$. Then $X\cong Y$ if and only if $I(X)\cong I(Y)$.

Question: In which way are the two different types of "classification theorems" related? Does a classification theorem of the first type imply a classification theorem of the second type or vice versa?