No, New York cops didn't have copper badges, buttons, helmets, or anything else. The word "cop" means to capture, seize, or snatch (first recorded use 1704), so one who cops is a copper. "Copper" as slang for a police is first recorded in George Matsell's Rogue's Lexicon (1859) under the entry for "copped", which meant arrested. "Cop" is a shortening of the word. All the early uses of the term given by the OED are British, except for Matsell, who was a New York Police Chief.