Landflap
There are many different types of tests,
none of them, to my knowledge, test for presence of "the virus" as in
whole, viable, infectious virus - that doesn't mean the tests are bad or wrong or that they detect common cold or that they "do not test for the presence of the virus at all".
RT-PCR, the most commonly used (I think) test and certainly the earliest available, tests for the RNA of the virus, identifying specific sections of viral RNA, unique to the virus. Some of the genetic markers used can trigger in lab conditions on the original SARS, none on common cold (see e.g.
https://www.fda.gov/media/136151/download performance evaluation section 2, analytical specificity). In real-world, no test is 100% but the PCR is >95% (usually much greater), so if you get a positive test there is a much less than 5% chance it's false.
RT-PCR on the other hand does have a high false-
negative rate, up to 30% in some studies. Significant part of this is down to the sample process, which isn't exactly easy or comfortable or exactly repeatable. So if you get a negative test, there is still a chance you have it, and if you have symptoms or significant risk recent contact, that chance may be quite high. In addition to that problem, PCR will only go positive several days after infection, -ve today can still be +ve tomorrow. This is why testing travellers on arrival is pretty pointless and no substitute for quarantine.
Where there is frequent confusion on PCR is that it
can show positive on
fragments of covid virus, left over debris from your body destroying it. So with a positive test you may not still be shedding viable virus, hence you may not be infectious - but still you almost definitely
had it.
Antibody and antigen tests that do do something almost not quite like "identify changes in immune system gene make-up" are a different thing altogether. These tests won't trigger for some weeks after you were infected and unless you are very ill in hospital you will almost certainly not be infectious by then. They are only really useful for seeing how many people in a population have had it. Also although >95% of infected people do make detectable antibodies, they aren't lasting long, tailing off significantly after 3 months or so. No one knows how long immunity lasts, there are T-cells as well, which last longer, and T-cell tests, but no one knows how much immunity these confer or if they reduce disease severity or actually prevent you catching it again and being a carrier/spreader. Which is why "antibody passports" are also pretty pointless and no substitute for quarantine.