Seeking for a counter-example for regularity properties of abelian categories

Can one construct a Grothendieck category which has enough projective objects, which is also locally finitely generated (or maybe even locally noetherian), but which does not have enough finitely generated projective objects?

More generally, a lot of questions on independence of some regularity/finiteness properties of abelian categories are very natural, but hard to answer. If anyone knows a somewhat systematic reference with (counter-)examples and other answers about this (a lot are spread in the literature, but the books or articles that I know on this topic contain only very few things and are focused on other aspects of abelian categories), I will be very interested!

Many thanks in advance.