The test for antibodies is a completely separate test.
The test they’re talking about here is for active viral proteins in the subject’s system. It means the virus is currently present. It doesn’t mean you are showing symptoms. A positive test can also come after the infection has been beaten since some viral proteins can remain for a period of time (typically days, but in some cases I’ve read about it being there for weeks with SARS-CoV-2), depending on how widespread the infection was.
The campaign is doing the responsible thing and regularly testing their workers. When some tested positive, they were removed to prevent further spread. My main concern would be President Trump, VP Pence, and any older members of the crowd. We need all of them healthy and ready to vote in November.