In the current theory of GR (which predicts the existence of black holes), black holes themselves are point singularities - so the only meaning for "size" is the event horizon.

Note that physicists don't believe this 0-size point is actually correct - the expectation is that a theory of quantum gravity would predict some fixed size for the black hole, one that also prevents the infinities inherent in the GR description of black holes.

Not quite. Point singularities can't spin, and since a black hole with no spin as about as possible as real mass with a temperature of absolute zero, every black hole we're ever likely to observe is actually a ringularity or other two-dimensional configuration.

Thanks for this comment, it was enlightening. I'd never much thought about this because physics is really not my forte, but after a few minutes of thinking about it, it's really intuitive why a point singularity can't spin. I can't explain it in any properly rigorous or technical way, but it makes no sense that someone could (by convention, of course) designate a "front" and "rear" to a mass with zero length in every direction.

What does it mean “to spin” in this case?

It means the black hole has angular momentum, which is detectable due to the resultant frame dragging (effect of the spinning on the spacetime around it) and presumably also due to the flattening of the sphere of the event horizon.

As I understand it, the fact that black holes are rotating precludes them from actually being a point singularity -- "rotation" having no meaning for a non-dimensional point. So they need to have some kind of disk or ring structure (still infinitely thin, however). (Note: I'm not a physicist, could be completely wrong).

That is the event horizon: when discussing a black hole's "size", you're discussing the event horizon.

Worth noting: despite the fact that this black hole would have an event horizon 4.5 cm in diameter, that doesn't mean you could get anywhere near it or put it in a box or anything.