I find it best to say that they're both, otherwise we're using the phrase that the KGB folk used, because that's the usual status of somebody who's quarantined.
The USSR had a long history of using "patient" to mean prisoner--this person is suffering from a mental illness, you can tell because of their wrong politics, and they must be placed in a psychiatric hospital until they are well. (When I hear people discussing conservativism or some other -ism as a mental illness, I just think of their spiritual and intellectual forebears. They're simply unaware of the violation of human and civil rights that they're advocating, adding ignorance to their foolishness.)
You quarantine people that aren't patients, as well. If you came into contact with an infected person and aren't yet in the testing window, you to get locked up.
The way you have to do quarantining to be as effective as possible is to identify those who are possibly infected. They are then under what amounts to forcible detention. They don't go home. They can call to make arrangements, but they don't get to go home to make sure the kids are okay or to pack things. And since they might be infected, might not be, you hold them separate from those known to be infected and from each other.
The Chinese way of doing things was to use clinical diagnosis, including CAT scans, before administering the COVID PCR test. This would exclude the asymptomatic. Of course, the false negative rate would also cut some infected with COVID loose. It made for a very leaky sort of quarantine, which led to a very harsh lock-down and the perpetual fear of contagion from outside as travel from outside needs to be curtailed or harshly controlled and the borders are a risk. The surveillance system that was set up for political infection works really well for SARS-CoV-2 infection detection.
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