I'm learning about Euler characteristic of orbifold, and struck by the statement that it can take fractional values. However, I've come across several definitions of orbifold Euler characteristic that don't look consistent with each other. For instance, from Sec. 2.4 in this paper, we have $$\chi(X/G) = \sum_{\sigma\in X} \frac{(-1)^{dim(\sigma)}}{|\Gamma(\sigma)|} \tag{1}$$ The orbifold I'm interested in here is the quotient space $X/G$, and the sum above is taken over all open cells $\sigma$ in the underlying space $X$. $\Gamma(\sigma)$, a subgroup of $G$, is the local group that leaves $\sigma$ invariant. With Eq. (1), I can show that $$\chi(X/G) = \chi(X) - \sum_i(1-\frac{1}{p_i})$$ for orbifold $X/G$ with cone points with order $p_i$. For orbifold $S^2/\mathbb{Z}_{2p}$ (modding out sphere $S^2$ by a $2p$-fold rotation), I get $\chi=1/p$, a fractional value that I've seen people talking about, nice!
However, I also see this definition from here, $$ \chi(X/G) = \frac{1}{|G|}\sum_{g \in G} \chi(X^g) \tag{2}$$ where $X^g$ is the set of points in $X$ left fixed by the action of $g$. With Eq. (2), I don't seem to get a fractional $\chi$ anymore. Again using $S^2/\mathbb{Z}_{2p}$ as an example, I get $ \chi=\frac{1}{2p}[\chi(S^2)+(2p-1)\cdot\chi(\text{\{2 points\}})]=2$. In fact, I have tried many other simple cases, like with $X$ being other 2D closed surface such as a torus, or a hollow ball in 3D, and then take $G$ to be $\mathbb{Z}_n$. For all those cases I tried, the definition in Eq. (2) always gives me $\chi \in \mathbb{Z}$.
Am I getting confused due to a stupid mistake somewhere? Or is it that Eq. (1) and (2) are simply two different definitions of orbifold Euler characteristic? Which one is better then? By the way, can one possibly get a fractional $\chi$ from Eq. (2)?
Finally, any comments that help me understand better the significance of a fractional Euler characteristic is appreciated. Thanks!