I have my employees dump (or hose down) water on the floor, put some cleaner on it. Swish it around with the mop. Let it sit for a minute. Then squeegee it to the drain. Then go back and mop it with clean water (with soap). It looks much better that way. When you do it like that the majority of the dirt ends up in the drain and not on the mop head or in the bucket. But they can’t really do that out in the store where the customers are.
I had one employee who spent the last year or two of her time with us on some sort of bizarre power trip making up rules. A few times I’d hear her yelling at employees for using “way too much water on the mop” and telling them to “wring it out more”. I could tell when she was mopping because I could hear the practically dry mop being dragged across the floor. The floor looked just as bad when she mopped as it did when others mopped with filthy water.