If you read the initial section or two of the chapter by Tate in Cassels and Frolich, he gives a nice explanation of how to pass from the classical formulation in terms of generalized ideal class groups w.r.t a modulus, and the more modern formulation in terms of idele class groups. As Tate explains,
the two formulations are indeed equivalent, but it is not quite as simple as saying that $I^c_K = C_K$. Here is a sketch of the equivalence (of course it is the same is in Akhil Mathew's answer, just slightly more detailed):
Since $I_K^c$ has already been taken to denote the ideals prime to $c$ (at least, this is how I interpret your notation), let me use $J_K$ to denote the ideles for
$K$. Then we can consider the subgroup $J_K^c$ of the ideles whose entries are all $1$ at any finite place dividing $c$, and at any infinite place.
Then there is a natural surjection $J_K^c \to I_K^c$ given by sending any element
$(a_{\wp})$ of the former to the ideal $\prod_{\wp} \wp^{v_{\wp}(a_{\wp})}$
of the latter (where the product is over finite places, i.e. prime ideals,
$\wp$).
Now one can show that $K^{\times} J_K^c$ is dense in $J_K$, so
the image of $J_K^c$ is dense in $C_K$. Since $N_{L/K}(C_L)$ is open in $C_K$,
we see that $J_K^c$ surjects onto $C_K/N_{L/K}(C_L)$.
Now one checks that this map factors through the surjection $J_K^c \to I_K^c$
described in the preceding paragraph, and in fact induces an isomorphism
$I_K^c/i(K_{c,1}) N_{L/K}(I_L^c) \buildrel \sim \over \longrightarrow
C_K/N_{L/K}(C_L)$, as required.
In practice, suppose you want to compute the Artin map on an element of $J_K$:
the algorithm is you first multiply by a principal idele so that the resulting element is in $J_K^c$ times $N_{L/K}(C_L)$. (You may not know exactly what this
group is, but its not hard to at least identify an open subgroup of it: for example, at any complex infinite place $v$ the norm map is surjective, at any real place $v$ the image of the norm map at least contains the positive reals, and at any finite place $\wp$ the image of the norm map will contain elements which are congruent to $1$ modulo the power of $\wp$ dividing the relevant modulus $c$.)
Now the Artin map on $J_K^c$ factors through the surjection $J_K^c \to I_K^c$,
and is computed on the target using Frobenius elements.
Indeed, this was the argument via which local class field theory was originally proved; one took a local extension, embedded it into a global context (so that the original local situation was realized as $L_{\wp}/K_{\wp}$ for some abelian
extension of number fields $L/K$), and then defined the Artin map via the above computation (which means concretely that one passes from the possibly ramified
situation at $\wp$ to a consideration just at the unramified primes, where everything is easily understood just in terms of ideals and Frobenius elements).
Of course, one then had to check that the resulting local Artin map was well-defined independent of the choice of "global context".
Nowadays, one can define the local Artin maps at all places (unramified or ramified) first. However, in generalizations to the non-abelian situation (i.e. local and global Langlands) one generally uses the old-fashioned technique of proving certain global results first, and then establishing the precise local results by passing to a well-chosen global context and reducing to a calculation at unramified primes. (This is a bit of an oversimplification, but I think it is correct in spirit.) So (if one has an eventual aim of understanding modern algebraic number theory and the Langlands program) it is well worth understanding the passage between the idelic and ideal-theoretic view-points on class field theory, and practicing how to use the algorithm described above.