There is no precise definition of a "classification theorem," so I don't think one can assert with mathematical certainty that a classification theorem of type 1 implies a classification theorem of type 2, or vice versa. As others have remarked, most "satisfactory" classification theorems enjoy both properties, so the question of whether one implies the other is perhaps not so important in practice.
Having said that, I think one can make a case that type 1 and type 2 are logically independent. For simplicity, let's assume that there are only countably many objects, so that we can think of $C$ as a set of binary strings. Think of $C$ as being partitioned into disjoint subsets $C_1, C_2, C_3, \ldots$ where each $C_i$ represents an isomorphism class of objects. I'm going to reformulate your two types of classification theorems in the language of computability theory. By a machine I will mean a Turing machine that halts on every input.
Type 1 Result. We write down a machine which accepts (i.e., outputs 1 for) exactly one element from each $C_i$ and rejects (i.e., outputs 0 for) everything else.
Type 2 Result. We write down a machine which takes two inputs $x$ and $y$; it gives an error message if either $x\notin C$ or $y\notin C$, and otherwise it produces a pair of outputs $I(x)$ and $I(y)$ with the property that $I(x) = I(y)$ if and only if $x$ is isomorphic to $y$. (Note carefully that I have written $I(x) = I(y)$ and not $I(x) \cong I(y)$.)
I'm not claiming that what I've called Type 1 and Type 2 results perfectly capture what you or anyone else means by a classification theorem, but I think it's a reasonable approximation, and it gives us a way to examine the logical relationship between Type 1 and Type 2.
A Type 1 result exhibits what one might call a choice function. If you have a Type 1 result, then you can list exactly one element from each isomorphism class, just by looping over all binary strings $x$, printing out $x$ if the machine accepts $x$ and printing out nothing if the machine rejects $x$. Furthermore, if someone hands you something that is allegedly on the list, you can use the machine to check if it really is on the list.
A Type 2 result provides what one might call a complete set of invariants $I(x)$ for any $x\in C$. To check whether two things are isomorphic, you compute the complete set of invariants in each case, and see if they are equal.
Having a Type 1 result does not automatically mean you have a Type 2 result. It is tempting to think that if $x\in C$, then we should be able to set $I(x)$ equal to the unique $y\in C$ that is isomorphic to $x$ and that the machine accepts, but the trouble is that there could be infinitely many elements that are isomorphic to $x$, and we do not necessarily have an algorithm to find which one of them the machine accepts.
Similarly, having a Type 2 result does not automatically mean you have a Type 1 result. A Type 2 result does in some sense let you "list" exactly one element from each isomorphism class, because you can loop over all binary strings $x$ and print it out if and only if $I(x)$ is a set of invariants that you have not already seen, but it does not automatically give you an algorithm for choosing a unique representative from each $C_i$.
I should point out, though, that if we demand a little bit more from a Type 2 result—namely, that $I(x)$ is actually something isomorphic to $x$ (so that $I(x)$ is a canonical representative and not just a complete set of invariants)—then a Type 2 result does automatically give a Type 1 result (at least if it is decidable whether $x\in C$ at all). Simply accept $x$ if and only if $I(x) = x$ (and $x\in C$).
