methinks you mean earliest or oldest common ancestor
Most recent common ancestor is correct - the oldest common ancestor is (possibly) an protein chain near a geothermal vent some 3.5 bya.
but they come to virtually the same thing:
Absolutely not.
would not the oldest common ancestor of all living genetically modern humans also be a genetically modern human?
Yes. But it is only a bound. It is not only possible, but highly likely that there were many generations before 'eve'.