Positive PCR is quite the opposite thing. It means you have an active virus in your system and it is probably reproducing. So, you are definitely not immune. Positive antibody test means that either you had a virus and recovered or you had been vaccinated and your organism developed antibodies.
The positive test proves that I HAD Covid (on May 20 at least). Here in Argentina I understand that 10 days after showing symptoms (or doing a positive test) and with mild symptoms you receive the OK (i.e. el alta médico). Meanwhile my PCR is negative and I am fine (also from a medical point of view, looking at the 10 days rule). My question evolves around where I could get a paper from doctor which exactly states that I am now "safe to interact with other people".
I now also checked the definition of Germany what is proof of immunity: proof that you had Covid (i.e. a positive PCR test) that is not older than 6 months but older than 28 days (
source). Other countries do not have exact definitions, at least I haven't found any others.
But I guess you are right that medical OK to leave the quarantine (i.e. ok to interact with other people) is not the same as proof of immunity, at least when looking at the example of Germany:
Day 0: positive PCR test (proven Covid infection)
Day 14: (or day 10 in Argentina) you can leave self isolation when having only mild or even no symptoms (alta epidemiologica / OK again to interact with other people)
Day 28: you have proof of immunity (at least in the sense that you are excluded from doing a quarantine on arrival)
Day 180: you lose proof of immunity
Currently I am - at least according to the German definitions - between medical OK, but do not have proof of immunity yet.