Instructor pilot reported communications problems with ATC and asked for clarification in approach clearance while flying over terrain below approach altitude.

Narrative

Instructional flight years ago with really good instrument student in stage 3 of a Part 141 program. Doing cross-country IFR flight planning from ZZZ to POU and student requested to do the VOR Runway 24 approach at POU. He wanted to work on VOR navigation and using 2 OBS for a non-precision approach. I thought it was a good idea until we had some trouble with ATC that day. I don't know if date is 100% correct; but it was close to Date; which is what made me think of it this year. We requested the approach and then the Controller was very busy confusing it with the VOR-A approach into POU. After attempting to clarify; he proceeded to give us radar vectors for the approach. Both the student and I were a little confused; but went with it despite out actual attitude being less than prescribed for each quadrant of the approach plate. ATC vectoring altitudes can be lower and we were VFR in some turbulence; 10+ SM visibility and clear with no ceilings. The reason for concern was that it is a hilly area and if 'Vectored to Final'; it is directly over the path of the ZZZ1 traffic pattern. There was also some confusion over this instruction as 99% of the time when you are 'Vectored-to-final' the 'intermediate' course is identical to the 'final' approach course with respect to heading/course changes/corrections. Did he want us to fly to the intermediate segment (course 218); intercept; and then remain on segment of published approach; or did he want us to fly to intercept 242 final approach course before that and continue with vectors? Upon looking this up in 7110 afterwards there seems to be some confusion and ambiguity with these terms when intermediate segment is not the same 'course'. Being VFR we continued as instructed; but if it were IMC this could be extremely problematic. We were low over unfamiliar terrain; irregular topography; another airport; in turbulence; and NOT on a segment of the published approach; above minimums; until passing the VOR into final. The flight continued there and back without incident and caused the student and I to look into ATC instructions; PIC compliance; instrument approaches; and safe practices during instrument procedures during our next ground lesson. I apologize for not including this form in the lesson during our ground debrief and recap of the situation. It could have been more accurate and useful had we done it with it fresh in our minds.