I was raised with the engine off. CB750. However, I've since been educated that Honda switched their method in priint, in their manual, to engine running, later in the series. Since the design of the CB750 tensioner never changed, it is my position that both work. MANY here were never exposed to the prior engine off instructions and adjusted it engine on and never had any problems.
It is my position that a CB750 chain cannot jump a tooth even at maximum slack on the adjuster.
I still prefer engine off.
No comments on other models.