Yes, you've covered it quite well. It's all about stability and learning how to stabilize the shoulders and lower back. In reference to the shoulders, even though the stabilizers are found in the front and back of the body, the focus is often in the, since that is where the shoulder blades are and we, as humans, tend to be forward focused. Going into shoulder stability requires some depth, but you can begin by concentrating on pressing the shoulderblades into the rib cage, keeping them as flat as you can, and holding them there while doing whatever pose you are in. I've realized that many people have this question so I plan to create a video on shoulder stabilization soon.
We do have plenty of videos on core stability and strength, though. Let me know if you would like me to recommend specific videos.