Why does a 3/8" bush cistern lever actually measure 6/8" in diameter?

[UserSteve]

Or does the 3/8" refer to the radius of the bush, not the diameter?

 

[oz-plumber]

It may be due to when the pipe sizes were determined (steel pipe) the internal bore of the pipe decided the pipe size.
Most, if not all steel pipe is measured ID.
So you have the wall thickness of the pipe which determined the thread size.