In chapter 13 in his notes on 3-manifolds, Thurston defines the orbifold fundamental group to be the group of deck transformations of the universal cover of the orbifold. He also makes a statement "Later we shall interpret $\pi_1(O)$ in terms of loops on $O$, but this interpretation doesn't seem to appear in his notes.

My question is, well, what is this interpretation, precisely?

Here are my thoughts so far:

The example I'm currently interested in is the 1-D orbifold S^{1}/ℤ_{2}. Its universal cover is ℝ. Deck transformations are generated by a translation by 2π, which I'll call T and a reflection about the origin R. There's relations $R^2=1$ and $TR=RT^{-1}$. ~~If I haven't messed up, this is the same as the fundamental group of the Klein bottle as well (if someone can explain how to construct the Klein bottle from S~~ Oops, I mixed up something in my head. I'll have another question on this in the future, perhaps.^{1}/ℤ_{2}, I would greatly appreciate it as well!). How can I relate paths on S^{1}/ℤ_{2} to loops in the Klein bottle?

I think my main trouble is making all of these observations precise, so a good reference with standard terminology / theorems (with lots of examples like the one I've been thinking about) would be appreciated as well.