I'll attempt to answer some of your questions. First off, 1) and 3) are equivalent. This is because the bicategory of fractions in 3) is the Morita bicategory of topological groupoids, which is equivalent to the bicategory of topological stacks. What maps are you inverting in 3)? Well, if you have an internal functor $F:G \to H$ of topological groupoids, you basically want to know when is this functor morally an equivalence, but I can be more precise. If you have a functor of categories, it is an equivalence if and only if it is essentially surjective and full and faithful. Note that this uses the axiom of choices, namely, that every epimorphism splits.
An internal functor $F:G \to H$ is a Morita equivalence if
1) it is essentially surjective in the following sense:
The canonical map $$t \circ pr_1:H_1 \times_{H_0} G_0 \to H_0$$
which sends a pair $(h,x),$ such that $h$ is an arrow with source $F(x),$ to the target of $h,$ is an surjective local homoemorphism 'etale surjection)
2) $G_1$ is homeomorphic to the pullback $(G_0 \times G_0) \times_{H_0 \times H_0} H_1,$ which is literally a diagramatic way of saying full and faithful.
You asked in 1), why 'etale surjection? Because this is what makes the map, when viewed as a map in in the topos $Sh(Top)$ an epimorphism, since the Grothendieck topology on topological spaces can be generated by surjective local homeomorphisms. If you want to use another Grothendieck topology (e.g. the compacty generated one, as I do in one of my papers), you must adjust accordingly.
Anyway, $F$ satisfies 1) and 2) if and only if the induced map between the associated topological stacks is an equivalence. Note though, that surjective local homeomorphisms don't always split (if they did, then every Morita equivalence would have an inverse internal functor). Hence, we have to use spans to represent morphisms, where one leg "ought to be" invertible.
Finally, you say that $G$ and $H$ are Morita equivalent if there is a diagram $G \leftarrow K \to H$ of Morita equivalences. This is the standard definition of Morita equivalent. Since the bicategory of fractions with respect to Morita equivalences is equivalent to topological stacks, one can also say that $G$ and $H$ are Morita equivalent of the have equivalent topological stacks.
Now, I'll respond to some of the comments:
@Zhen: If $G$ and $H$ are 'etale (or more generally 'etale complete) then they have equivalent classifying topoi if and only if they have equivalent stacks. In fact, there is an equivalence of bicategories between 'etale topological stacks, and the topoi which are classifying topoi of 'etale topological groupoids ('etendue). For more general topological groupoids however, there is information lost when passing to their classifying topoi.
@Ben: I believe this is related to Zhen's question. The definition in the way you stated it, usually appears in topos literature, and is related to the fact that open surjections of topoi are of effective descent. This is not a good concept when the groupoids in question are not etale. To deal with torsors one really wants to have some version of local sections.
One thing to keep in mind is that Bourbaki started in the 1930s, so in some sense simply too early to include category theory right from the start on, and foundational matters were rather fixed early on and then basically stayed like this. Since (I think) the aim was/is a coherent presentation (as opposed to merely a collection of several books in similar spirit) to change something like this 'at the root' should be a major issue.
Some 'add on' seems possible but just does not (yet) exist; and it seems the idea to write something like this was (perhaps is?) entertained (see below).
To support the above here is a quote from MacLane (taken from the French Wikipedia page on Bourbaki which contains a somewhat longer quote and source):
Categorical ideas might well have fitted in with the general program of Nicolas Bourbaki [...]. However, his first volume on the notion of mathematical structure was prepared in 1939 before the advent of categories. It chanced to use instead an elaborate notion of an échelle de structure which has proved too complex to be useful. Apparently as a result, Bourbaki never took to category theory. At one time, in 1954, I was invited to attend one of the private meetings of Bourbaki, perhaps in the expectation that I might advocate such matters. However, my facility in the French language was not sufficient to categorize Bourbaki.
There it is also mentioned that (in the context of the influence of the lack of categories on the discussion of homological algebra, only for modules not for abelian categories):
On peut lire dans une note de bas de page du livre d'Algèbre Commutative: « Voir la partie de ce Traité consacrée aux catégories, et, plus particulièrement, aux catégories abéliennes (en préparation) », mais les propos de MacLane qui précèdent laissent penser que ce livre « en préparation » ne sera jamais publié.
This translates to (my rough translation): One can read in a footnote of the book Commutative Algebra: "See the part of this Treatise dedicated to categories, and, more specificially, to abeliens categories (in preparation)", but the sentiments of Mac Lane expressed above [part of which I reproduced] let one think that this book "in preparation" will never be published.
The precise reference for the footnote according to Wikipedia is N. Bourbaki, Algèbre Commutative, chapitres 1 à 4, Springer, 2006, chap. I, p. 55.
Best Answer
The influence of Brandt's groupoid definition on the definition of category by Eilenberg and Mac Lane has been discussed on the category discussion list.
Bill Cockcroft told me in 1964-70 that there was an influence; he had visited Chicago for a year some time earlier. The use of groupoids in algebra was common knowledge in the 1940s, see the 1943 book on rings by Jacobson (N Carolina), and I expect the earlier book by AA Albert (Chicago), though I have not looked at that.
I did ask Eilenberg in 1985 about the influence of groupoids; he denied it and said that if it had they would have put it in as an example! I forgot to ask Mac Lane!
Papers [18,19] on my publication list (pdfs available) also have an extensive bibliography on groupoids.
Paper [147] "Three themes in the work of Charles Ehresmann: Local-to-global; Groupoids; Higher dimensions" gives an impression of Ehresmann's interest in geometric applications of groupoids.
Reidemeister's 1932 book on "Topologie" mentions the fundamental groupoid, and the groupoid determined by a group action. A recent translation to English by John Stillwell is available as arXiv:1402.3906.
The presentation in Galway explains my own interest in groupoids, through irritation that the usual van Kampen theorem did not compute the fundamental group of the circle, THE basic example in topology. I managed to find a solution to that in paper [4], using nonabelian cohomology, but the solution using the fundamental groupoid on a set of base points in [8], inspired by a paper of Philip Higgins, was more useful.