Why is a distribution defined in terms of the inequality
$$ |\langle\Gamma, \psi\rangle| \leq C \sum_{|\alpha| \leq N} \sup_{x \in S} | \partial^\alpha \psi |$$ for all $\psi \in C^\infty_c (\Omega)$, $S \subset \Omega$, $S$ compact, $K$ some constant, $\alpha$ a multi-index. And the existence of such constants $C$, $N$.
Now why is this definition written the way it is? Does it imply something about continuity of the distribution $\Gamma$ with respect to $\psi$? If so, what?
Obviously something like $$\Gamma (\psi) = \int_{\Omega} |f \psi| \, \, d \mu(x) $$ satisfies this if we let $$K = \mu(S)\int_{\Omega} |f| \, \, d \mu(x)$$ so I see why this definition is satisfied. But why is it necessary?
To answer this properly, one has to talk about the reason for introducing distributions first.
One of the main reasons for this is to be able to define a sort of generalized derivative for objects which are to "singular" to possess derivatives in a classical sense. For example, the functions
$$ f\left(x\right)=\begin{cases} 0, & x<0,\\ 1, & x\geq0 \end{cases} $$
has no derivative (at $0$) in the classical sense, but we will see below that it has one in the sense of distributions.
First, let us note (in dimension $1$ for simplicity), that if $f \in C^1(\Bbb{R})$ and $g \in C_c^\infty (\Bbb{R})$ (we could take any interval), then partial integration yields
$$ \int_\Bbb{R} f'(x) g(x) \, dx = f(b) g(b) - f(a) g(a) - \int_a^b f(x) g'(x) dx = -\int_\Bbb{R} f(x) g'(x) \, dx, $$
where $a < b$ are chosen such that ${\rm supp}(g) \subset (a,b)$.
The first idea is now to identify the function $f$ with the functional
$$ \varphi_f : C_c^\infty(\Bbb{R}) \to \Bbb{K}, g \mapsto \int_\Bbb{R} f(x) g(x) \,dx, $$
where $\Bbb{K} \in \{\Bbb{R}, \Bbb{C}\}$ is your favorite field. To do this, one should observe (this is sometimes called the fundamental lemma of the calculus of variations) that if $\varphi_f (g) = 0$ holds for all $g$ and a (continuous/locally integrable) function $f$, then $f \equiv 0$ (a.e.).
As the calculation above shows, we have $\varphi_{f'}(g) = - \varphi_f (g')$ for all $g$.
We note that the right hand side (that is, the map $g \mapsto \Phi (g')$ makes sense as soon as $\Phi$ is a linear functional on $C_c^\infty$ (for example $\Phi = \varphi_f$), so that we could define the "generalized derivative" of such a functional $\Phi$ by defining
$$\Phi' : C_c^\infty(\Bbb{R}) \to \Bbb{K}, g \mapsto - \Phi(g')$$
Observe that this is well-defined because of $g' \in C_c^\infty$ if $g \in C_c^\infty$.
The new idea now is to define a new class of generalized functions as the set of all (reasonable) linear functionals on $C_c^\infty$. As seen above, we will then be able to define a generalized derivative of these generalized functions such that it generalizes the ordinary derivative (i.e. $\varphi_f ' = \varphi_{f'}$ if $f \in C^1$).
But all we really wanted to do is to be able to define arbitrary derivatives of continuous/locally integrable functions. So it would really suffice to consider the class of all (generalized) derivatives (or arbitrary order) of generalized functions from the set $\{\varphi_f \mid f \in C^0(\Bbb{R}) \}$ (or replace $C^0$ by $L^1_{\rm loc}$). Instead of these "nice" objects, the class of all linear functionals on $C_c^\infty$ (in contrast to the continuous ones) would contain many pathological functionals which we want to avoid.
Instead of introducing this definition, we observe that each of the functionals $\varphi_f$ is continuous in the sense described in your question (as you observed yourself). We also observe that this continuity property is inherited by the generalized derivative, i.e. if $\Phi$ is continuous in the sense described above, then $\Phi ' $ is also a continuous linear functional in this sense (why?).
One can even show (this is done in Rudin's "Functional Analysis") that this definition yields the (more or less) smallest class of "generalized" functions such that it contains all continuous functions and arbitrary derivatives of those. More precisely, for each compact set $K \subset \Omega$ and each distribution $\Phi$, there is some $N \in \Bbb{N}$ and continuous functions $f_\alpha$ such that
$$ \Phi(f) = \sum_{|\alpha| \leq N}\int_\Omega f_\alpha \partial^\alpha f(x) \, dx $$
holds for all $g \in C_c^\infty(\Omega)$ with support in $K$.
This observation justifies the choice of functionals made above.
Also note that there is a general principle at work here: We want to have a "big" space of generalized functions. To achieve this, we choose a very "small" space ($C_c^\infty(\Omega)$) with a very strong topology/notion of convergence, i.e.
$$ f_n \to f \text{ in the sense of } C_c^\infty (\Omega) $$
if and only if there is a compact set $K \subset \Omega$ such that ${\rm supp}(f_n) \subset K$ for all $n$ and $\partial^\alpha f_n \to \partial^\alpha f$ uniformly for all $\alpha$ and define our "large" space to be the dual space of the "small" space, i.e. the set of all continuous linear functionals.