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Because of Hawking Radiation, black holes are actually not entirely dark, and they have a temperature. Just because it's in the name doesn't mean it's part of the definition. Think of it as "black" as in "black-body radiation".

This is true, but Hawking radiation has nothing to do with the observed event. Hawking radiation is incredibly weak, orders of magnitude dimmer than even the cosmic microwave background, never mind visible starlight. The observed event is as the parent poster said, originating from just outside the hole's event horizon.

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