I am studying orbifold fundamental group (or more generally orbifold homotopy groups). In a nutshell, my questions is: what are they intuitively? In what follows I give definitions and more precise questions. My definition of orbifold fundamental group is via classifying space of groupoid, which is explained in the next paragraph (so you may want to skip it if you know the definition).

Let $\mathcal{G}$ be a topological groupoid consisting a topological spaces $G_{0}$ of $objects $ and $G_{1}$ of $arrows$ together with usual continuous structure maps. Let $|\mathcal{G}|$ denote the associated topological space $G_{0}/G_{1}$. Let $G_{n}$ be the iterated fibered product $G_{n}=G_{1}\times_{s,t} G_{n-1}$. These $G_{n}$ have the structure of a simplicial topological space, called the $nerve$ of $\mathcal{G}$. Face operads $d_{i}:G_{n}\rightarrow G_{n-1}$ for $i=0,\dots,n$ are given by $$ d_{i}(g_{1},\dots,g_{n})=(g_{1},\dots,g_{i}g_{i+1},\dots,g_{n}) $$ for $i=1,\dots,n-1$ and $$ d_{0}(g_{2},\dots,g_{n})=(g_{2},\dots,g_{n}), \ \ d_{n}(g_{1},\dots,g_{n})=(g_{2},\dots,g_{n-1}). $$ The classifying space $B\mathcal{G}$ of $\mathcal{G}$ is then defined as $$ B\mathcal{G}=\bigsqcup_{n}(G_{n}\times \Delta^{n})((d_{i}(g),x)\sim(g,\delta_{i}(x))), $$ where $\Delta^{n}$ is the topological $n$-simplex and $\delta_{i}:\Delta^{n-1}\rightarrow \Delta^{n}$ is the standard facemap.\

The $n$-th orbifold homotopy group of $\mathcal{G}$ based at $x\in |\mathcal{G}|$ is defined to be $$ \pi_{n}^{orb}(\mathcal{G},x)=\pi_{n}(B\mathcal{G},y), $$ where $y\in G_{0}$ maps to $x$ under the quotient map $G_{0}\rightarrow |\mathcal{G}|$.

The following are my questions:

Why is this a reasonable definition? Any manifold $M$ can be thought of topological groupoid via its chart i.e. $G_{0}=\bigsqcup_{i}U_{i}$ and $G_{1}=\bigsqcup_{i,j}U_{i}\times_{M} U_{j}$. It is not clear to me that the definition above reproduce $\pi_{n}(M)$.

I am aware of an explicit description the orbifold fundamental groups of the orbifold Riemann surface $\Sigma_{g,n,k}$ of genus $g$ and $n$ orbifold points $p_{i}$ of order $k_{i}$: $$ \pi_{n}^{orb}(\Sigma_{g,n,k}) =\langle \alpha_{i},\beta_{i},\sigma_{j} \ (1\le i \le g,1\le j \le n)\ | \ \sigma_{1}\dots\sigma_{n}\prod_{i=1}^{g}[\alpha_{i},\beta_{i}]=1,,\sigma_{i}^{k_{i}}=1\rangle $$ Is it easy to see this explicit presentation by the definition above?

It seems there are several ways to define the fundamental group of an orbifold, such as covering space etc. How should one understand orbifold fundamental groups?

Thank you for your assistance.