Edit: I'm not going into detail in this comment because I am not an immunologist and I cannot express clearly and correctly what the difference is, but just be aware that "Immunity" may not mean 100% protection in the way that the general public thinks that word means.
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.
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'd be curious to see what your various definitions are.