Maybe I have misinterpreted these two definitions, so maybe someone could please enlighten me. Isn't Flabiness for a sheaf equivalent to it being generated by its global sections? I have heard of the latter in algebraic geometry when discussing about sheaf of modules over a ringed topological space, but I suppose the idea of sheaf generated by global section can be extended to most categories. Well one should imply the other, and I would say (if I didnt misinterpret anything) that Flabiness implies the latter. If they are not equivalent the question can be extended as follows:
In which categories are they equivalent? What are some simple examples of sheaves that are generated by global sections but are not flabby?