How to choose which origin vertex to snap from? In the picture below I am moving the higlighted cube to the left along y and have closest vertex snapping enabled. This is in object mode. As you can see, Blender has circled the vertex that it is snapping to but has automatically selected the top left vertex of the translating cube as the vertex from which to snap.

How can I instead select the vertex pointed at by the red arrow to be the one that is snapping to the circled destination vertex? Sometimes this seems to change automatically, but I can't find a setting or any documentation on how to do this.
Although I do not know exactly how to do it the way you are asking, I do know of a way that should make it snap to the correct vertex.
Move the highlighted cube to the far left of the other object.  When you choose to snap then, it will choose the closest vertex, which would then be the right one, as said by @Robin Betts.