It is hard to tell what technically really happened and where the HSA reporting was inaccurate.  Perhaps it would be wise that they always run the story by someone with piloting and/or aircraft maintenance background to ensure it is on track.  That being said, perhaps the flight instructor could have done a better job (plus, will be talked to by the FAA as to why he turned this student loose for a cross-country flight with questionable skills) explaining emergency procedures and dos and don'ts to his student.  The last thing you want to do is re-initiate a takeoff after a hard landing which may have damaged the airframe/engine/flight controls and could be catastrophic.   It also raises issues about judgement and common sense that should be inherent in any pilot.  I have always said anyone can learn to fly if they have adequate funds available, at least a reasonable amount of coordination (not a total klutz), fair amount of determination/passion, and sufficient critical thinking.