If circles are aligned on the red or green axis, the vertical circle is the path. When circles share 2 axes, it makes a difference which one is selected for the path.
Beyond that, I’ve found the “rules” to be just a pit different depending on if circles intersect on all three axis or only two. I don’t think some of these “rules” are noted in the videos and other online help items that I found before posting. This is something I did to help cement the ideas in my own brain, and I’m adding to the post to help future newbies like myself in understanding how easy spheres can be.
I’ve just started with Sketchup and learned a lot from this thread. Great conversation, and thanks everyone for the input.