So then, what does "common descent" mean, if not some kind of relationship? I sense a semantic shell game here.
This is semantics at all - it's very straight forward.
"Common descent" means that two species share a common ancestor. This is the same principal of two cousins sharing a common ancestor - their grandparents.
The relationship of humans and modern apes is the same that cousins would have to each other - they both share a common ancestor.
I am descended from my mother and father.
My sister is descended from my mother and father.
My sister is not descended from me, and I am not descended from her, but we are related.
I really cannot think of simpler words to explain the concept.