For the circles part: it's true.

Assume that each circle defined by our set of points $S$ contains $\geq 4$ points; in particular, there are $\geq 4$ points in total. Fix one of our them, say $p$, and consider all the circles determined by our points and passing through $p$; each such circle passes through $\geq3$ points from $S-\{p\}$.

Now apply inversion with center $p$ - under this, the above observation shows that $S-\{p\}$ goes to a set of points with the property that any line through two of them contains a third. Applying the standard version of the theorem, we see that all the images of $S-\{p\}$ under the inversion are collinear - hence, $S-\{p\}$ all lie on a circle through $p$.

For the circles part: it's true.

Assume that each circle defined by our set of points $S$ contains $\geq 4$ points; in particular, there are $\geq 4$ points in total. Fix one of them, say $p$, and consider all the circles determined by our points and passing through $p$; each such circle passes through $\geq3$ points from $S-\{p\}$.

Now apply inversion with center $p$ - under this, the above observation shows that $S-\{p\}$ goes to a set of points with the property that any line through two of them contains a third. Applying the standard version of the theorem, we see that all the images of $S-\{p\}$ under the inversion are collinear - hence, $S-\{p\}$ all lie on a circle through $p$.