Sorry this is incomplete, but "Natural Immunity" means a person's immune system will mount an effective response to an infectious agent. However, that person may still be infected, and may be contagious to some degree while the body is fighting off the contagion.

I'm not a doctor or biologist, but I've been learning over the past year. For instance, people generally think of things like "infected", "sick", and "contagious" as being the same thing (or nearly the same thing). If you're talking about one person in everyday context, that's fine. If you're talking in a clinical or epidemiological context, you have to be more careful.

I’m still learning, but the concept of “sterilizing immunity” has started to come up recently. What I infer from cursory reading is that “immunity” means the body will mount an immune response immediately upon exposure to the virus, but it doesn’t mean that an infection won’t take hold. “Sterilizing immunity” means immunity strong enough to prevent the virus from growing at all.

The issue at hand is that those who have been vaccinated and/or recovered from COVID seem to be able to contract the disease again, even to the point where they can be infectious to others, even if they don’t get terribly ill or even show symptoms.

If I understand correctly, it means that the immune system launches an immediate response to the new infection, it doesn't mean that the immune system successfully defeats the new infection, just makes it more likely. And then there's the problem of variants.