Most of the homological algebra books that I've checked develop chain complexes in the context of modules over a ring. The two exceptions I know would be Schapira notes "Categories and homological algebra" and the section 12.13 on stacks project.
I wanted to ask if you can point me to other references (es. notes) that develop them in the context of abelian categories, possibly giving more details than the aforementioned. My knowledge of abelian categories comes from Borceux Categorical Algebra II, so for example I would expect the definition of homology to be something similar to the following (ie. making use of exactness and additivity):
I understand that there is no particular merit of writing down those results without using elements, I ask this as personal linguistic preference.