Winterize my lawn mower
The "article" you linked to is written by the manufacturer of STA-BIL, a brand of fuel stabilizer. So I would take the entire thing with an enormous grain of salt.

I don't see how you can go wrong following the advice of the lawn mower manufactorer. Personally for my small engines I switch to stabilized fuel sometime in the fall so that it works its way through all the hoses and into the engine, then when I'm done for the year I run it dry. That way virtually all the fuel is gone and whatever is left should have stabilizer in it.