I have often said that "training the core" has developed into just another form of isolation.
I believe in full-body movements that may (or may not) emphasize one aspect or attribute more that another while retaining full-body power. This is one of the key aspects of the RKC and the Hard-Style School of Strength.
Example 1: The military press. What is the difference between the "shoulder press" and a proper Hard-Style military press?
Full-body tension. Total tension in heavy lifts equals safety and improved perfromance. Grip the ground with your toes. Tighten the legs, lock the knees, squeeze the glutes, pull up the pelvic floor, compress the ribs, compress the breath and squeeze the weight as you press. You WILL be stronger.
Example 2: Hanging leg raise. Try a HLR and see if you can do it WITHOUT tensing the arms and legs (which incidentally, ain't part of the core). Again, Full-body tension is needed.
Example 3: The beloved Pushup. Not the sway back, chicken head pushup most people try to get away with. Do a pushup with a 3 second descent, 2 second pause and explode back up. Full-body tension.
Example 4: Any and all grip work. Try to close a hard CoC gripper, bend a nail, tear a phone book, farmers walk and what will you need?.....I'll take "Full-body tension" for $500, Alex!
Example 5: We mustn't forget the snatch! Try heavy KB snatches in multiple sets of low reps with short rest periods. You obliques will thank you.
Example 6: The deadlift....are you starting to get the picture?
Your body is a single unit, not a collection of parts. Train it that way.