First, a trivial remark. Any object $c$ of a category $C$ is both the colimit and the limit of the functor $c \colon 1 \longrightarrow C$ that sends the unique object of the terminal category $1$ to $c$. So in particular, the category $\mathbf{2} = (0\to 1)$ is both the colimit and the limit of the functor $\mathbf{2} \colon 1 \longrightarrow \mathbf{\text{Cat}}$.
Now, more interesting, as you suggest in your question, is when we can build up an object as a colimit of simpler objects. We can see the category $\mathbf{2}$ as a weighted colimit in $\mathbf{Cat}$ in a number of ways, for example:
as the tensor $\mathbf{2}\ast 1$, i.e. the colimit of the functor $1 \colon 1 \longrightarrow \mathbf{\text{Cat}}$ weighted by the functor $\mathbf{2} \colon 1 \longrightarrow \mathbf{\text{Cat}}$;
as the coinserter of the parallel pair $0,1 \colon 1 \longrightarrow 2$ (here $2$ is the discrete category with objects $0$ and $1$), which is the colimit weighted by the parallel pair $0,1\colon 1 \longrightarrow \mathbf{2}$;
as the co-comma category
$$
\require{AMScd}
\begin{CD}
1 @>1>> 1 \\
@V1VV \Longrightarrow @VV1V \\
1 @>>0> \mathbf{2}
\end{CD}
$$
i.e. the colimit of the span $1 \longleftarrow 1 \longrightarrow 1$ weighted by the cospan $0 \colon 1 \longrightarrow \mathbf{2} \longleftarrow 1 \colon 1$. In fact this follows from the previous example, since co-comma categories can in general be constructed by first taking a coproduct ($1+1 \cong 2$) and then taking a coinserter (as above);
as the colimit of the arrow $1 \longrightarrow 1$ (seen as a functor $\mathbf{2} \longrightarrow \mathbf{\text{Cat}}$) weighted by the arrow $1 \colon 1 \longrightarrow \mathbf{2}$. (For reasons to be hinted at below, this kind of weighted colimit is called a "lax colimit of an arrow".)
Hence the category $\mathbf{2}$ can be seen in a number of ways as a weighted colimit of discrete category (i.e. set)-valued diagrams, and moreover of diagrams with constant value $1$. However this is in some sense not entirely satisfying, since $\mathbf{2}$ already features in the weights of all these colimits. We can remedy this concern by considering lax colimits in $\mathbf{\text{Cat}}$.
The lax colimit of a functor $F \colon A \longrightarrow \mathbf{\text{Cat}}$ is a category $\texttt{laxcolim}F$ together with a lax natural transformation $F \longrightarrow \Delta (\texttt{laxcolim}F)$, such that for each category $C$, the induced functor
$$[\texttt{laxcolim}F,C] \longrightarrow \text{Lax}[A,\mathbf{\textbf{Cat}}](F,\Delta C)$$ is an isomorphism. Here the domain is the category of functors $\texttt{laxcolim}F \longrightarrow C$ and natural transformations between them, and the codomain is the category of lax natural transformations $F \longrightarrow \Delta C$ and modifications between them.
By spelling out the definition, one can see that any category $A$ is the lax colimit of the functor $\Delta 1 \colon A \longrightarrow \mathbf{\text{Cat}}$ with constant value $1$. In particular, the category $\mathbf{2}$ is the lax colimit of the constant functor $\Delta 1 \colon \mathbf{2}\longrightarrow \mathbf{\text{Cat}}$, also known as the lax colimit of the arrow $1 \longrightarrow 1$.
(Note that there is a general construction by which a lax colimit can be calculated as a weighted colimit; for our description of $\mathbf{2}$ as a lax colimit, this construction retrieves the weighted colimit of example 4.)
