> I would say "early 2000"

"Early 2000" is between January and March/April that year. August is clearly shortly after exact middle (June/July).

> or "around 2000"

This would mean some years before or after year 2000, not middle of the year 2000 (aka "mid-2000").

> Mid-2000 sounds like around 2005 to me.

2005 would be "mid-2000s". Note the trailing 's'.