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) do it this way, as does Hellen Colman (and so do I, see chapter 2). I gather the idea goes back to Haefliger.

Really orbifolds are objects of a bicategory (see Lerman's paper 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.

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) do it this way, as does Hellen Colman (and so do I, see chapter 2). I gather the idea goes back to Haefliger.

Really orbifolds are objects of a bicategory (see Lerman's paper 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.

EDIT: To answer the second part of question 1, the geometric realisation of the groupoid associated to the open cover of a manifold is (at least) homeomorphic to the manifold, hence the fundamental group computed the two different ways are isomorphic. To see this via the description I give, one has to know that the 2-functor $\pi_1\colon OrbifoldGpd_* \to Grp$ factors through the bicategory of pointed orbifolds as constructed by e.g. Lerman (and many many others), i.e. sends Morita/weak equivalences to isomorphisms of groups. The open cover groupoid is weakly equivalent to the manifold thought of as a groupoid, and the description of the fundamental groupoid I give, in the case of a manifold (or space) is naturally isomorphic to the usual description.

To give a pointer for question 3, Moerdijk and Mrcun cover this in their chapter in this book, although possibly using sheaves instead of covering spaces.

I claim (in answer to the first part of question 1) that this (the definition given in the question) is a reasonable definition precisely because of its relation to the definition I give. I should probably discuss things like the homotopy type of topological stacks (see work by Noohi) and the relation between topological groupoids and topological stacks, various methods for computing with these and so on. But if the first paragraph is incomprehensible, then perhaps the long version will be as well :-)