The "core" is not a particularly well defined group of muscles, but it would commonly include the following muscles.
This is the "6 pack" muscle, often just referred to as "the abs".
Role: Spinal flexion (bending forwards at the lower back)
Isolation exercises: Most ab exercises, including sit-ups, planks, ...
The main purpose of the abs is not to do situps but to stabilize the spine.
The spine is a bit like an antenna mast.
The antenna mast have guy-wires to stabilize it against external forces such as wind:
The spine also have guy-wires in the form of muscles running from the base of the spine: the pelvis to the upper part of the spine:
The body can adjust the ...
In order to save your back you want to train your abs isometrically.
Goblet squats train the rectus abdominis very well.
Push-ups are also good.
Single arm kettlebell deadlift trains the obliques very well.
Zercher squats are also supposed to train the rectus abdominis + obliques very well.