It seems that there are many definitions of quasi coherent sheaves(modules). There is a nice page on nLab quasi coherent sheaves

My questions are:

Are there any other definitions of quasi coherent sheaves? For what?(I mean in which situation, they should be redefined to satisfy the working environment)

How are they related?

For question 2, nLab says these definitions are equivalent. There are four definitions in nLab. It seems except first definitions, the other three are equivalent, just restate in different language: i.e presheaves <---> pseudo functors <---> fibered categories.

But are these definitions **exactly the same as the classical definitions?** (as locally presentable modules)

big flashing red warning sign. There is an independent issue regarding coherent sheaves, and that is that when Hartshorne was writing his book he went for the easy "f.g. module" definition, which he knew wasn't Grothendieck's definition (and he says as much). His definition coincides with Grothendieck's (the "correct" definition) when the base is Noetherian, but not otherwise. It's a theorem (well, a lemma) in EGA that O_X (structure sheaf of a scheme) is Noetherian! This issue is independent of all this pseudo functor twaddle. $\endgroup$ – Kevin Buzzard Feb 14 '10 at 8:29