Just a short post to point out that I just defined the notions of “coarse quotient”, “good quotient”, and “geometric quotient” in the stacks project. To see the details see the chapter on Quotients of Groupoids. Please let me know if you think the definitions aren’t what you think they should be.
Really the idea for this chapter is to have silly things like: “U reduced => X reduced if X is a categorical quotient of s, t : R —> U” (from GIT, paragraph 2, Chapter 0) proved in excruciating detail and in ridiculous generality. It seems to me this is somewhat worthwhile since the first thing you always read about this material is that chapter of GIT and it can be confusing.
Also, the level of generality where R is not necessarily a groupoid can be useful, for example when forming the MRC quotient of a variety U you look at families T <--- C ---> U of rational curves C_t on U and the pre-relation you consider is R = C \times_T C (which has two maps to U). There is absolutely no reason that this should define a transitive relation and of course in general it really doesn’t.