It's in the Standardized Textbook of Barbering.

First pass with lather if the customer asks for the famous "Once-Over."

Second pass, against the grain, can be done with water from a bottle if the customer asks for a "Close Shave." Also, a second stropping is performed on the razor before beginning the second pass.

The clean-up mentioned in the original note is not a dry shave, but rather just a second pass.

A true dry shave would be dry skin with no prep. If you think this might work, all you have to do is try it once, and you will never do it again.