Reading across Milne's notes on Class Field theory, I came across the following definition on page 165. Here $S$ is a finite set of non-archimedean primes (of a number field $K$), and $I^S$ is the group of ideals in $K$ generated by the primes not in $S$.
Now for example take $K = \mathbb{Q}$ and $S$ as the empty set. For any given prime $p$ define $\chi_p : I \to \mathbb{C}^{\times}$ where $\chi(p) = -1$ and $1$ everywhere else. It is easy to see that according to the given definition $\chi_p$ falls into the same equivalence class for all $p$. But now as per the definition, all such $\chi_p$ and the trivial character satisfies the conditions for the primitive character in this equivalence class, violating the claim that there exists a unique primitive character.
In general, for any number field $K$ and any such equivalence class $C$ of characters we can define a primitive character as follows. Pick any random character $\chi: I^S \to \mathbb{C}^{\times} \in C$. Now we may define a character $\chi'$ such that $\chi'|_S = \chi$ and on $S$ in any arbitrary way since the primes freely generate the group of ideals. Such a $\chi'$ qualifies as a primitive character as per the definition. But clearly not unique.
There seems to be some misunderstanding on my side regarding the definitions. It would be really useful if someone could help clear my doubts.