(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
These two definitions are indeed equivalent.
By the discussion here the existence of finite limits is equivalent to that of finite products and equalizers. But an equalizer of two morphisms and in an Ab-enriched category is the same as a kernel of . Dually for finite colimits, coequalizers and cokernels.
In particular, is a kernel iff and dually.
Every morphism in a pre-abelian category has a canonical decomposition
Of course every abelian category is pre-abelian.
The category of torsion-free abelian groups is reflective in all of Ab. Therefore, it is a complete and cocomplete -enriched category, and therefore in particular pre-abelian. However, it is not abelian; the monomorphism is not a kernel.
The concept “pre-abelian category” is part of a sequence of concepts of additive and abelian categories.