Cops are civilians.

Not sure I follow. Cops ARE civilians in normal government run precincts. The general public gets a bit confused sometimes as it is muddied but the paramilitary structure.

Since police officers are not members of the armed forces (army, navy and so on) they are by definition civilian and are treated as such in a court of law.

They are sworn officers which gives certain additional responsibilities. But that would be the same for a government run police dept or a private dept.

While private police are rare in the US they do exist.. usually not entire departments but rather subcontracted officers often in specialized roles.