Seems pretty clear to me.
The sling rendered the big man unconscious, the sword killed him.
You sit are correct. Even if you just read it, without analyzing the word definition (see above), in context it only makes sense that he felled Goliath with the stone, and killed him with the sword. Digging deeper will still provide more evidence of that. The answer is simple, he knocked him out (prevailing in the contest), and cut off his head (killing and taking his trophy).