Yes, copper is safe to cook on. The darker area around the edge of that plate is not an "original coating," it's just the area where the copper has tarnished over time.
The reason the copper has a "bare" look in the center is most likely because the cooktop has been maintained with a cleaning product specially intended to remove the tarnish. This is the appropriate way to maintain and use copper cookware. While copper work surfaces can be kept bare and allowed to tarnish, or intentionally colored through the application of e.g. particular acids and then sealed with a clear epoxy, copper cookware should be maintained with appropriate cleansers to keep the surface bright.
(NOTE: When I say "work surface" I'm talking about something that is not heated and usually doesn't come into direct contact with food. Commonly called a kitchen counter (AmE), countertop (AmE), worktop (BrE), or bench (AuE/NZE). When I say "cooktop" I'm referring to an appliance that produces heat for cooking, and when I say "cookware" I'm referring to the pots, pans, etc. that actually come into contact with the food.)
Natural and chemical weathering of copper surfaces can create a variety of different compounds, some of them dangerous. The only significant risk I'm aware of with naturally-weathered copper cookware is verdigris, the distinctive green patina seen on e.g. the Statue of Liberty. I manufactured my own copper countertops about ten years ago and did some research on safety to decide whether to seal the surface with an epoxy or let it weather naturally; I found only one notable incident of deadly verdigris poisoning, related to industrial cooking vats that were not cleaned and which developed a heavy patina that then leached into the food.
It's not difficult to remove patina from copper; anything acidic will usually do the trick, even ketchup. (Of course, you would wipe the ketchup away and dispose of it after using it to dissolve potentially toxic copper compounds. Better just to use a cleaner specifically designed for copper cookware. I do not recommend the readily available Bar Keeper's Friend products as they're designed more for SS and will produce a black byproduct on copper.)
All this said, I wouldn't recommend cooking on bare copper surfaces. They're a lot of work to maintain, we have better solutions in modern cookware that use copper as a conductive layer married with other metals for the cooking surface, and there is always the chance of an adverse reaction to the small amount of copper that does end up in the food.
