It's natural to think that the correct analogue of the groups $({\mathbf Z}/m{\mathbf Z})^\times$ in a number field $K$ should be the groups $({\mathcal O}_K/{\mathfrak a})^\times$ where $\mathfrak a$ is a nonzero ideal in $\mathcal O_K$, and for some purposes that is true. But for other purposes it is not, and one instance where it is not is your question.
Hecke's motivation for creating "his" characters was to produce $L$-functions of them as Euler products over (nonzero) prime ideals in $\mathcal O_K$ that generalize Dirichlet $L$-functions. If you start off with a character $\chi$ on a unit group $({\mathcal O}_K/{\mathfrak a})^\times$, how do you make it into a function of ideals? You want some kind of series like
$$
\sum_{\mathfrak a} \frac{\chi({\mathfrak a})}{{\rm N}(\mathfrak a)^s}
$$
running over (nonzero) integral ideals ${\mathfrak a}$ of ${\mathcal O}_K$, and if $K$ has class number greater than 1, it's hard to imagine how to get a function of ideals out of a function on $({\mathcal O}_K/{\mathfrak a})^\times$. Even if $K$ has class number 1 you'd have pretty serious problems making such a transition if there are units of infinite order in $\mathcal O_K$, which there are except when $K$ is ${\mathbf Q}$ or an imaginary quadratic field.
The key to understanding how Hecke generalized Dirichlet characters is to reinterpret the group $({\mathbf Z}/m{\mathbf Z})^\times$ as the multiplicative quotient group of fractional ideals $I_{(m)\infty}/P_{(m)\infty}$, not as a group of units in a quotient ring. That leads to generalized ideal class groups $I_{\mathfrak m}/P_{\mathfrak m}$ for a generalized modulus $\mathfrak m$ in a number field, and it is characters of generalized ideal class groups $I_{\mathfrak m}/P_{\mathfrak m}$, not characters of the groups $(\mathcal O_K/\mathfrak a)^\times$, that are examples of Hecke's definition of his characters. The characters of generalized ideal class groups all have finite order, but Hecke's definition is much broader: it allows for infinite-order characters that are not closely related to any finite order characters in any way. Generalized ideal class groups are the original way in which class field theory was developed, and you're not going to find anyone telling you that the formalism of class field theory is easy to grasp the first time through it.
Hecke's original definition of his characters did not make any use of ideles, which in fact weren't created until later (by Chevalley). His paper introducing his characters came in two parts in Mathematische Zeitschrift (vol. 1 in 1918 and vol. 6 in 1920) and he gives explicit examples of his characters for real and imaginary quadratic fields. The classical definition is discussed on the Wikipedia page you link to in your question, although the definition there is (at the moment) all largely in words and is kind of opaque. I think you would find the classical formulas defining Hecke characters in general pretty frightening. You can find them in, for instance, Narkiewicz's book on algebraic number theory. Hecke's original papers do not offer much in the way of gentle motivation for his definition. These developments, at the time, were not at all obvious. In the 1940s, Matchett showed in her thesis how to interpret Hecke's characters more conceptually as the characters of the idele class group, and that is often how they are viewed today because it is a cleaner and more conceptual definition.
