I am not an electrical engineer, but I am a computer engineer (Master's from RIT) and that involved about 1/3 EE, 1/3 Computer Science, and 1/3 its own classes. Certainly there are many who know more than I do, but I'm not wholly ignorant of what I am saying...or at least attempting to say. It's very possible that what I think, and am attempting to write, is not quite making it into words very clearly.
Power is defined as voltage times current (at any given instant, or an average integrated over time); I have no argument there, and of course the definition of power is not Ohm's law.
Ohm's law relates voltage applied to something to current that flows through it, making the voltage proportional to the current with the resistance being the constant of proportionality. Many materials behave in this way in practical circuits, and so may be modeled as a resistor and analyzed using Ohm's law. This implies that the V/I curve for whatever being described is precisely linear, and passes through the origin. If the current through some device goes up when voltage goes down, then rather clearly Ohm's law does not apply to it, or at the least the resistance (or equivalent resistance) is not constant but is itself varying with the applied voltage in some fashion. For electric motors, the relationship of voltage and current also varies with the mechanical load applied to the motor, the speed the motor is turning at, and maybe some other things.
A Thevenin equivalent circuit is a very handy analysis tool for linear circuits. It doesn't apply to circuits with components that have nonlinear behaviors, such as semiconductors, although often an adequate model may be derived by assuming linear response over some limited operating range.
A silicon diode, for example, is hardly linear in its response, but can be modeled reasonably accurately by a pair of Thevenin equivalent circuits: a quite large impedance when reverse biased (the applied voltage is less than the threshold voltage of about 0.7V), and a small impedance with a voltage source equal to the threshold voltage when it's forward biased.