I agree with you that the plank is a great abdominal (not necessarily core) exercise. I frequently recommend it to clients.

However, it is really only the abdominals working - not the spinal extensors, as you're trying to resist extension - so the rectus abdominis is working. If you were trying to resist flexion, then the spinal extensors would be working.

The other thing that isn't 100% correct is that the high plank (aka pushup position) is a core exercise. It's not. When shifting from the elbow plank to the pushup position, the emphasis also shifts from the abdominals to the shoulders and pecs. If you were to hold the elbow plank as long as possible, you'd stop because your abdominals would give out. If you were to hold the high plank as long as possible, you'd stop because the pecs and anterior delts would give out.

Nothing wrong with that, but they're not progressions of the same exercise - they're different exercises.

Same with the one arm/opposite leg plank: it's a pec/delt exercise more than it is a core/abdominal exercise. Again, nothing wrong with that, it's just important to know the anatomy of what different exercises are working.