Latent as to their own illness.
This is all as far as I know as a lay person:
People who test positive for SARS-CoV-2, (the virus) may never become ill with Covid-19, but are expected to clear the infection and stop testing positive after a relatively short time, if they don’t become ill. They are people who have, essentially asymptomatic infection, which their bodies fight off, and then they are no longer infected, and can no longer spread the disease.
Others who test positive will have symptomatic Covid-19. There are multiple outcomes of that.
My understanding is, with HIV, once a person tests positive, without medication, it was extremely rare for a person to spontaneously clear HIV from their system – to recover from the infection. The person might remain healthy for a decade or more, but would continue to test positive the whole time. Eventually, they would become sick and die – uncontrolled HIV was almost always fatal. All of this is different now if the person has access to modern anti-HIV drugs, of course.
So, a person who tests positive for SARS-CoV-2 may or may not develop symptoms of Covid-19, but if they do it would be expected to happen in days. They may never develop symptoms, and clear the infection relatively quickly – asymptomatic infection. Whereas a person who tests positive for HIV would, untreated, be expected to continue testing positive for HIV indefinitely and would eventually become ill with AIDS. The latency period of the infection could last for a decade or more.