How is it identified? Is it a vertex? A special vertex one appoints? Is it something else? Is it just an arbitrary position member variable m_pivotPoint? What is it, how do we get it, how do we make it?

the pivot usually starts life as the center of your object bounding box… then you move it;

if u keep it as a separate (as u call it ‘arbitrary’) position than the center of your object then when u move the pivot, remember to also transform the center (or recalculate it) if the object is transformed in pivot space;

That's it … all the best ?

Pivot point transformation is the object space transformation of 0,0,0 point of object space, meaning identity transformation of pivot point will result in 0,0,0 point of object space, which is the world space position if in world space. You can center this pivot point to AABS, AABB or move it to whatever arbitrary point.