Core Strength = The ability to maintain neutral spinal/pelvic position under load and through dynamic ranges of motion. This is why "core strength" is so vital to waterskiing, we take tremendous loads from the boat and have to dynamically transmit that load through the ski using various ranges of motion. If you cannot maintain a neutral spinal/pelvic position then you will lose force or, worse, get hurt trying to produce force form a bad position. Glutes, hamstrings, and quads (to a lesser extent) are all vital parts to core strength because they all support and stabilize the pelvis.
If you are going to move a load in a squat that is 80-100% of your max the most important thing you have to do is maintain a neutral spinal position so that you can transmit the force to the prime movers efficiently. This requires a tremendous amount of activation from your core (erectors, abdominals, psoas, quadratus, as well as the deeper spinal supporters like multifidi and rotators). Thus squats are probably the best "core exercise" you can do because it uses the core in the way it will be used in life... to support and stabilize the spine/pelvis through a dynamic range of motion and under load. Deadlifts and olympic lifting are very good for this too.