Let us review some idea. If $f \in L^{1}_{\mathrm{loc}}(\Bbb{R}^n)$, then we can define an element $T_{f} \in \mathcal{D}'(\Bbb{R}^n)$ by
$$ \left< T_{f}, \phi \right> = \int_{\Bbb{R}^n} f\phi, \quad \forall \phi \in C_{c}^{\infty}(\Bbb{R}^n). $$
Then the embedding $f \mapsto T_{f}$ allows us to identify $L^{1}_{\mathrm{loc}}(\Bbb{R}^n)$ as a subspace of $\mathcal{D}'(\Bbb{R}^n)$.
Of course we may not write $u = u(x) = \cdots$ for arbitrary $u \in \mathcal{D}'(\Bbb{R}^n)$. But let us take a deeper look on the identification above. We observe that the value of $f$ at $x = x_0$ is completely determined by the family of functions $f \phi$ such that $\phi$ is supported on a neighborhood of $x_0$. Then every claim on local property of $f$ can be translated in terms of the local data $\{ f \phi \}$. For example, the value of $f$ at $x = x_0$ is $y_0$ whenever
$$ \lim_{\varepsilon \to 0} \left< T_{f}, \phi_{\varepsilon} \right> = y_0$$
for $ \phi_{\varepsilon}(x) = \varepsilon^{-n} \phi\left( \varepsilon^{-1} (x - x_0) \right)$, where $\phi \in C^{\infty}_{c}(\Bbb{R}^n)$ supported on a neighborhood of $0$ and its total mass satisfies $\int_{\Bbb{R}^n} \phi = 1$. In this way, we can regard $f = T_{f}$ as a function-like object such that at each point $x$ there corresponds a local data of $f$. The term generalized function is used for this point of view.
The key point is that, this idea extends naturally to distributions as well. In particular, $u$ is completely determined by its local data at each point $x \in \Bbb{R}^n$. But in this case, the local data no longer need not yield a value of $u$ at $x_0$. This is why we define the support of a distribution to be a subset of $\Bbb{R}^n$.
Of course, it can happen that the local data give rise to the function value of $u$ at some points. Indeed, we can speak of the value of $u \in \mathcal{D}'(\Bbb{R}^n)$ at a point $x_0$ whenever the condition above holds with $u$ instead of $T_f$. Thus it may also happen that $u$ reduces to a function on some open set. A good example is the Dirac delta $\delta_0$, where $\delta_0 = 0$ on $\Bbb{R}^n \setminus \{0\}$.
I will try to start from the notion of support of a function and obtain the definitions above in a natural way.
If $f : \mathbb{R}^n \to R$ then its support is defined as $S = \overline{\{x \in \mathbb{R}^n : f(x) \neq 0\}} $ For the purpose of discussion it's easier to talk about $S^c$ instead of $S$, namely $S^c$ is the largest open set where $f = 0$.
So far, so good, but distributions are not functions, so it doesn't make sense to say that the value of a distribution at a point is $0, -1, \pi$, etc. However, distributions are linear functionals, so it's not unreasonable to define that a distribution $T$ is zero on an open set $\omega$ if it "doesn't do anything there". In other words, for an arbitrary $\phi$ smooth, compactly supported in $\omega$ then $\langle T, \phi \rangle = 0$. Thus, we have arrived at the definition of open annihilation set that you mentioned.
Now, to define the support of $T$ we take the complement of the largest open set where $T$ vanishes: just like in the case for support of a function $f$: look at the disussion about $S$ and $S^c$ above.
I hope this helps.
Note: it's worth checking that if $T$ is induced by a (locally) integrable function $f$ in the standard way, then the support of $T$ will be the support of $f$, in other words the two definitions are consistent.
Best Answer
Definition: Let $u \in \mathcal{D}'(\Omega)$, where $\Omega$ is a open subset of $\mathbb{R}^{n}$. We say that $u$ is smooth near of $x_0$ if there exists a open neighbourhood $\omega$ of $x_0$ sucht that $u|_{\omega} \in C^\infty(\Omega)$.
Definition: Let $u \in \mathcal{D}'(\Omega)$, we say that $u \in C^\infty(\Omega)$ if there exists $f \in C^\infty(\Omega)$ such that $T_f=u$, where $T$ is the canonic injection of $C^\infty$ in $\mathcal{D}'(\Omega)$.
Definition: The singular support of a distribution $u \in \mathcal{D}'(\Omega)$ is defined by $\operatorname{sing supp} u=\Omega \setminus\{x \in \Omega: u \hbox{ is smooth near of } x \}$.
It's easy to see that $\operatorname{sing supp} u$ is the complementary set of biggest subset of $\Omega$ in which $u \in C^\infty$.