The short answer is, because I don't have time to write a full answer, is that the fundamental group is composed of equivalence classes of formal composites of paths and elements of the arrow space of the corresponding groupoid. If $p\to[0,1]$ is a cover by closed intervals only overlapping on boundaries (i.e. a partition), then such a formal composite is a functor $p\to G$. Geometric realisation makes this an element of your definition. Moerdijk and Mrcun (see e.g. [this](http://www.math.psu.edu/mehta/slides/dblgpds-cornell.pdf)) do it this way, as does [Hellen Colman](http://arxiv.org/abs/math/0612257) (and [so do I](http://ncatlab.org/nlab/show/Fundamental+Bigroupoids+and+2-Covering+Spaces), see chapter 2). I gather the idea goes back to Haefliger. 

Really orbifolds are objects of a bicategory (see [Lerman's paper](http://arxiv.org/abs/0806.4160/) on this), and this bicategory can be described using anafunctors, and anafunctors from $[0,1]$ to an orbifold are equivalent to what I described above.