How do I translate this propeller start procedure into English?
If the engine is already started, and if this is a turboprop with a constant speed propeller, it is most likely that the pilot is performing a pitch control check.  In this case the pilot would move the throttle into Beta, or reverse, which would momentarily cause the propeller to "push".  After this he would come back to idle, where if the propeller control lever was set to fine pitch or max RPM, the prop would spin "free".  The actual procedure and intercom voice calls will vary by aircraft and operator.      

This answer presumes that he is not yelling out a window, or transmitting with aircraft callsign to a ground controller.  A video clip showing where he is on the ground, whether the prop is turning, and what his hands are doing at the time of the calls would confirm this.

If it is difficult to post a clip here, just watch what his right hand is doing:  If he appears to disengage a latch and pull the main power lever backwards, (you may hear an audible change in the engine sound as the propeller reverses pitch) and declares “propeller pushing”, then moves the lever back forward again and declares “propeller free” this would be pretty strong confirmation of my interpretation.

If that is the case, I might suggest a literal translation as you described in the question.  If you know the exact make and model of aircraft you could maybe narrow down what an English speaking pilot would say instead, but the literal translation wouldn’t necessarily be wrong either.