I'll write some more about this later.
For the moment, I'll make a few remarks: "an orbifold is a topological space which looks locally like a finite quotient of Rn by a finite group of O(n)" is a bad definition. For example, it fails to address the (important!) class of non-effective orbifolds. E.g. the moduli space of elliptic curves is a non-effective orbifold because every elliptic curve has a $-1$ automorphism.
Second: it is important to realize that orbifolds form a 2-category, not a category! If you get something that looks like it's a category, you're doing something wrong.