How does the power distribution change for a turbofan engine from being stationary to going at full speed?
Your definitions are correct and your derivations correctly use the law of conservation of energy, so yes, the power distribution change does happen.

The more important question is what does this actually mean. Energy is a rather curious quantity in that while it is conserved in each reference frame (including non-inertial as long as appropriate potential is included), the values of kinetic and potential energy will be different in each!

You can look at all the cases in the reference frame of the aircraft and then the propulsive power will always be zero, because the velocity is—by definition of the reference frame—zero. But the very same situation viewed from the reference frame of the Earth it will be as you describe, and from the reference frame of the air mass it will be similar, but the values will differ slightly as the reference frame move relative to each other at the speed of the wind.

Note that enthalpy does not vary in coordinate transformation, so the sum of the power towards moving the aircraft and towards accelerating the air stream is will be the same in all the reference frames. Most easily calculated in the reference frame of the aircraft from difference in kinetic energy between the intake and the exhaust. This is the output power of the engine itself, and comparing this with the heating value of the fuel will give you the thermodynamic efficiency. Thermodynamic efficiency of turbine engines increases with flight speed due to pressure recovery, i.e. the ram air pressure increasing the effective compression ratio.

Also note that this variance in propulsive power is there for all kinds of propulsion systems, including propellers and wheels of ground vehicles. The propulsive power is always zero at standstill, and how quickly it increases depends on the reactive mass they have available. The more reactive mass, the larger the propulsive efficiency (= propulsive power to output power) will be, and the faster it will increase with velocity. Propellers have some more reaction mass available due to larger diameter, and wheels have whole Earth available, but their thrust is still limited by the surface friction and the maximum torque of the engine (and the friction of the clutch for engines that don't produce torque at zero RPM).