I actually disagree that charing it while off will "help extend battery"
The reason I disagree is that the charger provided with the phone will output a max current. If the phone is on, some of that current goes into power the phone, while the rest (minus some efficiency) goes into the batteries.
So if the phone is on, the battery will charge slower (and this also makes sense)
Now, most people know that charging a battery quickly is bad for the battery. If you charge a battery too fast, it can heat up past a safe point - which will doubtfully ruin the battery, but will surly decrease the batteries life-span (not charge, but overall 1-2 year lifespan)
It makes more sense to charge the battery slowly, allowing the chemicals inside to react in a slow, stable way