This is exactly the sort of issue that I was thinking about. Ultimately it is the presence of the antibodies that matters, irrespective of how they got there I think. And yes, neither an infection, nor a vaccine necessarily guarantees anything. It is all a percentages game, and how long it lasts is another thing that we don't know for sure in case of COVID-19 yet.
Not quite, as far as I understand it. Antibodies do show your immune system would react quickly to a new infection, but not necessarily quickly enough to prevent a case of the disease. (Bad news, and perhaps why people can get Covid-19 more than once.) On the other hand, the immune system can remember antigens and build a very quick response when stimulated by an antigen, thus creating immunity, even when there are no antibodies present in the normal, unstimulated state. IIUC, immunity to many diseases, whether induced by the disease or by a vaccine, works this way.
Also, the immune system works by detecting some foreign protein or genetic material (antigen) produced by the germ (virus, bacteria, or other parasite) and remembering it for future infections. The antigen or antigens selected by the immune system are not necessarily the same as the ones contained in a vaccine, and sometimes the antigens in a vaccine are MORE effective or produce longer lasting effects than natural immunity. It depends on the virus and the vaccine. In other words, the immunity from a vaccine might last less time or longer than the immunity from the virus. The problem is at this time, we just don't know.
We also don't know if immunity (from the virus or from one of the vaccines) always wears off in a few months, or if it usually lasts for years but in some people or under some conditions, wears off quickly, or if the re-infection cases are actually false positives in either the initial or second case, or the person never actually recovered and still had an active infection even after appearing to recover. The virus could hide somewhere, like chicken pox, and then re-emerge months later. This is another unknown. But after almost a year and millions of cases, there should be enough repeat cases to draw some limits. The answers should emerge in a few months to some of these questions. It will take anywhere from 6 months to several years to know how long vaccine-induced immunity lasts.
The good news about both the Pfizer and Moderna vaccines is the 90-95% effectiveness comes from actually observing the rate of infection in double blind tests. The people who got the real vaccine showed 5 to 10% of the number of cases as the equal number of people who got the placebo, and the number of cases in the people who got the placebo is close to the expected number for unvaccinated people, so there isn't some hidden selection effect. (For example, there weren't far more insusceptible people in the experimental group than in the general population.)
Another item of good news is that even if the immunity is only short-lived, at 90%, with good penetration, it should knock down infection levels low enough that contract tracing and isolation of exposed people would be able to stamp out any outbreaks in the long run. Better, cheaper, faster testing would help enormously with this, but only if the infection rate is low enough.
I recently heard of a home test that works be smearing a nasal swab (Q-tip) onto a specially treated strip of paper and observing if it changes color. It costs about 30 cents, and takes less than a minute. If this is true (it could be totally bogus or a ridiculous exaggeration), then it would make contact tracing and isolation much easier. If you were exposed to infected person X, someone will come to your door with a box of a weeks worth of strips and tell you to isolate and test everyone once a day. If no one is positive in a week, you're all clear. If anyone tests positive then call the 800 number. And give us a list of everyone you've been in close contact with since you were exposed to person X. This kind of contact tracing really only works if there are a very small number of infected people, but it is extremely effective.
Very few people are immune to Yersinia pestis, there is no vaccine for it, it is endemic in animals in many parts of the world but very few people die of bubonic plague any more because of good tracing and effective treatments. There is hope!