Why does the Moon's terminator look "wrong" in this image?
The reason that moon image looks wrong is because it is wrong. It is not a real image of the moon $-$ at least the terminator is not real.

The original article you cite has a link just below their image indicating the source of their image of the moon. That source is the night sky planner, hosted by JPL. You'll find the same image on that website, albeit slightly darker (it seems the NPR people lightened up the image a bit).

If you do some more digging, you'll see that the night sky planner got its image of that moon from someone else. Within the html code, they have the image defined as:

&lt;img alt="the moon" src="http://api.usno.navy.mil/imagery/moon.png"&gt;


Clearly you can see that the image was taken from the United States National Observatory.

After a bit of digging, I found out what exactly is going on here. The purpose of this site is to show you the current phase of the moon. To do this, they take a single image of the full moon and artificially shade out a region to make it appear as the current phase of the moon. You can see their process here and how it was done by some guy named R. Schmidt. They've broken down the moon phases into 181 images which you can download here, if you're interested.

As you can see, the terminator on that image is wrong because it is not a real image of the current phase of the moon, but rather a computer generated "shading" of the full moon to indicate the current moon phase.