You're right except the exercises you mentioned really don't have much to do with the abs main function. That's just flexion.
Rotary stability is one of the main jobs and stability in general. The type of exercises to do are all kinds of planks, rotational movements like cable woodchops or russian twists, barbell russian twists, side bends like Saxon side bends, anti-rotation like Pallof press, AR barbell, for flexion (minor) ab-pulldowns and Janda situps will be a better choice.
I see that people understand that doing supplemental "core" work is very important. But what we now have to get is that CRUNCHES AIN'T CORE WORK.