What the American system (circa 1820-30) discounts is that the older British system (circa 1650-1660) is counting through the steps of the relationship not the generations.
There are only 2 steps to a grandparent, i.e. subject to parent 1st step, parent to grandparent 2nd step.
There are also 2 steps to an aunt or uncle, i.e. subject to parent 1st step, parent to sibling (aunt,uncle), 2nd step. In other words the term Aunt or Uncle replaces the term Grand.
The next step (3rd step) logically is therefore the greats; great grandfather, great grandmother, great uncle, great aunt.
In addition DNA (centimorgan chart ) has actually displayed the British system is the more correct way of describing such relationships.
Cheers
Guy