Orbifold fundamental group in terms of loops?

I guess that the easiest way to think of this is to consider a disc with a singular point $x$ of index $2$. Take a loop $\gamma$ that turns once around $x$. Then in the universal cover of the disc, $\gamma$ lifts to a path $\tilde\gamma$ whose endpoints do not coincide (they are the two lifts $y_1,y_2$ of $\gamma(0)$). You cannot homotope it to a constant while keeping its endpoints over $\gamma(0)$ (which you would need to get an homotopy of $\gamma$ with fixed base point). Therefore this loop is not trivial.

Consider now $2\gamma$. Then in the universal cover, it lifts to a loop (that goes from $y_1$ to $y_2$, then from $y_2$ to $y_1$, continuing to turn around the (unique) lift of $x$. But now this universal cover is a disc (without singular point) and you can homotope your loop to a constant (based on $y_1$ with the above notations). This homotopy projects to an homotopy of $2\gamma$, which is zero in the fundamental group.

Most of the standard intro sources on orbifolds discuss their fundamental groups in terms of coverings. One exception is Ratcliffe's book "Foundations of Hyperbolic Manifolds", chapter 13 of which contains a discussion of the fundamental group of an orbifold defined via loops.

To understand the difficulties inherent in forming a "path-based" definition of the orbifold fundamental group, it is good to ponder Serre's definition of a fundamental group of a graph of groups, given in his book "Trees".