Is anyone able to explain why a turbojet is known as a thrust-producer, but a piston-prop combo is known as a power-producer?
A turbojet engine is called a “thrust producer” because it produces thrust directly. By this I mean that it does not use any other devices to convert the engine output into thrust.
The output of a piston engine is not thrust, but takes the form of a torque applied to the output shaft. The power transmitted through a rotating shaft is proportional to the torque multiplied by the RPM. So the output of a piston engine is rotating power. We call it Brake Horsepower or BHP.
To convert BHP into thrust we must use a propeller. The efficiency with which this conversion process is achieved varies with factors such as RPM, blade angle and TAS, so there is no fixed relationship between the BHP and the thrust produced.
Why, however, is it that the best angle of climb in a propeller aircraft is derived from minimum power rather than minimum drag?
Best angle of climb for any aircraft occurs at the speed at which the excess thrust is a maximum. This is true for both jet aircraft and propeller aircraft.
JET AIRCRAFT
For ATPL study purposes we assume that jet thrust is approximately constant at all airspeeds. This means that maximum excess thrust and best angle of climb will occur at Vmd.
PROPELLER AIRCRAFT
The thrust produced by propellers is maximum before the brakes are released at the start of the take-off run. Thrust then decreases as TAS gradually increases. When we compare this gradually decreasing thrust with the drag curve we find that maximum excess thrust occurs at a speed that is slightly less than Vmp. The difference between Vx prop and Vmp is very small, so some texts state that they are the same speed. But strictly speaking best angle of climb still occurs at the speed at which excess thrust is a maximum.
In reality all of the above is further complicated by things like constant speed props, but these can be ignored for the purposes of your question.