In your first definition V is the voltage supplied from the generator, which is the voltage across the wire AND the voltage across the load. Solving for I gives you the correct current to use in P=I^2R.
In your second definition V is still the voltage supplied to the load and wire. That would...
They should all be equivalent. Take Ohm's Law (V=IR). If you plug in V from Ohm's Law into your first equation you get P=I^2R. If you take Ohm's Law in the form I=V/R and plug it into your first equation you get P=V^2/R. I would guess the difference was a result of the problem setup, i.e...