I think you've pretty much gotten it.
My thinking is:
3 points determine a circle.
Once you have the circle, you can find its center (x) and radius (r). The plane of this circle is perpendicular to the line connecting the center of the sphere (X) to the center of the circle (x).
Use the sphere radius (R) and (r) in a right triangle to find the distance
between point x and the center of the sphere (X). R is the hypotenuse.
There is an ambiguity between 2 possible spheres - one to the left of the circle and one to the right.