The extinction of a person is defined as the extinction of that person's conscious existence forever. Your personal extinction is you never experiencing being consciousness ever again.
The word extinction is usually applied to a species of plants or animals. But an extinct species could exist again some where else in the universe if the right conditions of evolution occurred again. Or if a conscious beings made the correct sequences of DNA and then let them develop into reproducing organisms.
Finite singular theories of conscious existence believe that you become "extinct" when you die. But itomultiplicity theories create a scenario more like that with extinct species. Under the right circumstances, random or deliberate, there you are again experiencing conscious existence--- you being conscious.