The term 'human extinction' has two interpretations.
Read here for context.
One interpretation refers to the inevitable extinction of homo sapiens sapiens in geological time, the other, to the current major pruning of earth's biota.
Better terms have been coined to describe the latter, this current rapid whittling of extant flora and fauna.
- The sixth extinction.
This acknowledges that there have already been five major bottlenecks in biodiversity during recorded geological history. The phrase is also the title of a book on the subject by Richard Leakey
- The Late Quaternary extinction.
A phrase I first saw used in an article by the science writer David Quammen. He defined it as a hundred year period beginning 1914, and continuing until 2014. The final date probably dark humor based on trends he perceived at the time of writing.
The latter phrase is more resonant with the names for the other five extinctions as it is defined by a geological epoch and so has the same flavor in usage.
One reason this current extinction event is unique, is that it doesn't fall in line with a cause that has been claimed to tie together the other five.