I'm trying to compare the two following definitions of $C^*$-system trying to translate the one into the other.
Definition 1:
A $C^*-$dynamical system is a triple $(\mathfrak{A},G,\alpha)$ where $\mathfrak{A}$ is a $C^*$-algebra, $G$ a locally compact group, and $\alpha:G \rightarrow Aut(\mathfrak{A})$ a group homomorphism, which is assumed to be strongly continuous, i.e. for any $a \in \mathfrak{A}$ the map $g \mapsto \alpha(g)(a) \in \mathfrak{A}$ is norm continuous.
Definition 2:
A $C^*$-dynamical system is a pair $(\mathfrak{A},\Phi^t)$ $(t \in \mathbb{R})$ where $\mathfrak{A}$ is a unital $C^*-$algebra and $\Phi^t$ a one-parameter strongly continuous group of $*$-automorphisms of $\mathfrak{A}$.
In my understanding the second definition is given in order to encode the evolution of the system, so basically $G = (\mathbb{R},+)$ and therefore we are considering
\begin{align} \Phi : \mathbb{R} &\longrightarrow Aut(\mathfrak{A})\\ t &\longmapsto \Phi^t(a) \quad \forall a \in \mathfrak{A} \end{align}
that is assumed continuous in the strong operator topology, that is, $\|\Phi_\alpha(a)-\Phi(a)\| \rightarrow 0$ for each $a \in \mathfrak{A}$.
Also, choosing $\mathbb{R}$ as $G$ I'm automatically saying that the group $\{\Phi^t\}_{t \in \mathbb{R}}$ describes a REVERSIBLE dynamics? While choosing for example $\mathbb{N}$ means that my dynamics is dissipative?
Definition 2 is just a special case of Definition 1, where $G=\mathbb R$.
As far as your last questions are concerned, I'm not familiar with what is meant by the terms reversible and dissipative in this context, but if reversible is to mean that every $\alpha(g)$ has an inverse of the form $\alpha(h)$, then every $C^*$-dynamical system described by definition 1 is reverisble, as $G$ is a group and $\alpha$ is a group homomorphism.