Which engine manual must a pilot legally use?
FAR Part 91, in a certificated plane, private piot in a private plane, you can fly however you'd like.  Charles Lindberg taught lean of peak to pilots on WWII to save fuel for long distance ocean flights.

Check the engine manual, you will find you can fly as lean of peak as you can manage.  I find it hard to do with carburated O-360 engine. Usually I lean out until it gets rough, then enrichen some.  I just had a 4-channel engine monitor installed, so now I can start experimenting and see how well it really does LOP, maybe playing with some carb heat to help stir up the air flow better.
With fuel injection engines, you can adjust/swap out the injectors to get better fuel flow into each cylinder for better lean adjusting.

Leaning helps maintain good fuel/air mixture so you don't run too rich and waste fuel.  If you check later year Manuals, you may find better direction on leaning for your engine as well.  For the Cessna Cardinal, I have a 1977 POH that I use to supplement my plane's 1973 POH, Cessna got smarter as time went on it seems and some things are covered better.

Flying commercially may be a different manner. There the Company may specify one thing over another, you may not have a choice.