Stayability = This is a posed fixed position, there is not very much transfer to real world athletic movements. There are not very many sports where you stay there. Artificial and sterile.
Stability = This represents an instant in time, it is not a position or a posture, it is definitely not a still picture. A javelin thrower or a swimmer needs great shoulder stability, the ability to maintain integrity of the joint at high velocity and under imposed stress. In some ways it is a moving target. A good sound strength training programs that is proprioceptively demanding, works multiple joints and is tri-planar will address stability.