The first comment to this post points out that, given two (topological or localic) groupoids, they can be non-Morita-equivalent evenif their classifying topoi (topoi of equivariant sheaves) are equivalent.
Now, I thought that the equivalence of the classifying topoi was exactly the definition of Morita equivalence in the case of localic groupoids, so I'm confused by this remark. For example, consider the first comment to this other post: doesn't this agree with my definition?
Thank you in advance to anyone that can solve this ambiguity.
The accepted answer to the second post you've linked defines a "Morita equivalence" to be a particular kind of internal functor. This notion is also discussed in item (3) of that question. Say that two localic groupoids are "Morita equivalent" if there is a zigzag of Morita equivalences between them.
If two localic groupoids are Morita equivalent then they have equivalent classifying toposes. But, in general, two localic groupoids can have equivalent classifying toposes without being connected by a zigzag of Morita equivalences. We can fix this disconnect by restricting to étale-complete localic groupoids: if two étale-complete localic groupoids have equivalent classifying toposes then they are connected by a zigzag of Morita equivalences.