The definition of the core can be confusing as it sounds like the abdominals but it is more than that. The central core of our being is situated in our torso which is directly connected to many stabilizer muscles situated from our glutes to our necks. But the primary objective of any good core is to protect the spine.
Creating a specific core fitness workout should include the strengthening of some, if not all of these complex stabilizer muscles that we have all over our torso and connecting limbs. Core fitness would mean endurance training, whether that be strength endurance on these specific muscle groups like the internal and external obliques or general fitness conditioning from doing cardio.
The strength training of the rectus abdominis and the erector spinae are the agonist and the antagonist in many core movements but not all. The use of muscles like the multifidus and the gluteus maximus and minimus are just a few of the core muscles that would need to be isolated when doing a core fitness workout.
Instead of specifically isolating these body-parts, both strength and fitness can be achieved by doing super-sets between movements like squats and lunges. But that is only one choice of many other alternatives when working a core fitness routine. You cannot lift any weight without using your core.
If every weight that you lift works directly or indirectly on your torso or core muscles then it makes sense that the more weight that you can lift using strict form the stronger your core will be. As mentioned above the fitness would be both the strength and the cardio endurance needed for running or rowing.
The body’s ability to lift a weight or perform any explosive function is directly linked to the ability of the core to protect the spine. Simple movements like a push-up or V sits are all perfect for strengthening the core. There are many great martial art movements that have been specifically designed to work on the core like the dragon fly done by Bruce Lee.
Other movements specifically for the core are the back bridge or the plank which can all be more advanced like doing a plank on a ball etc. There are also side plank and ‘supermans’ that can be done to specifically isolate muscles of the core. But developing core strength is a different kind of training that developing core fitness.