I'm not familiar with the terminology or requirements in the US so it's difficult to comment.
In the UK, earth, ground and neutral are all the same thing, though used differently. That is don't go connecting them to the same point locally.
The reason you need to earth the lathe itself, and indeed the cabinet it sits on, is in case a live wire for the motor or switches breaks through its insulation and touches the lathe itself. Without an earth of some description you would provide a nice path when you next touched the machine.
Not nice

With the machine properly earthed the fuse would blow instantly.
So it it important to earth any equipment including any steel benches etc in you workshop.
It may just prevent you or somebody else from dying
