What's the smallest area a square can have that a cube can still be wrapped with it?

I am going to summarize the trio's solution here since it's essentially an answer to my question, but I'm hoping that the first paragraph in particular might be clarified by one of you as I don't completely follow. The problem, titled "A Cubical Gift" [10716], was posed in the 1999 volume of the Monthly and phrased differently ("What is the largest cubical present that can be completely wrapped (without cutting) by a unit square of wrapping paper?") Clearly, an answer to this question will solve mine and vice-versa.

Two arbitrary points in the square are chosen and their distance is considered before and after wrapping the cube with the square. They argue that the surface distance between the two points after wrapping is no larger than what it was before wrapping, since the paper was neither cut nor streched. [I'm not convinced. Perhaps the surface distance notion is throwing me off. Is it the distance along the paper? Wouldn't that then be an obvious statement, a tautology even? Otherwise, are we talking about the distance along the cube? Isn't there a lot of choices to call the distance in that case? And how does this paragraph relate to the next?]

Consider an arbitrary point on the cube. There exists another at least twice an edge length away [surface distance, again]. This implies that for any point on the square, another point can be found at distance at least twice an edge length. So, this is also true for the center of the square, implying that the diagonal is at least 4 times the length of an edge. Therefore, the side length of the paper square is at least $2\sqrt{2}$, giving the area of 8. Finally they show that the inequality is tight by demonstrating the folding pattern I described in the OP.