C172 Flight Instructor and Center Controller debate the need for a holding pattern at HAIGS when cleared direct and for the approach to Runway 27 at HGR.

Narrative

While proceeding towards HGR to conduct an ILS approach for training (the ILS Runway 27); the ZDC Controller directed us to proceed direct to HAIGS Intersection (e.g.; not a vector); and stated we were cleared for the ILS approach. We were then instructed to contact the HGR tower. Upon reaching HAIGS; IAW AIM 5-4-9(b)(3); I told my student to perform one turn in the hold in order to get established; following the published hold. As such; we performed a direct entry (we were arriving from the EMI VOR; roughly the EMI 326 Radial); and continued the approach after the one turn. We were under IFR in scattered IMC. Once established; the Tower Controller stated that they had received a call from ZDC indicating that I; in effect; should have proceeded straight in. I respectfully told the Controller that I did not agree; and that I was required to perform the one turn. Later; after performing the missed approach and contacting the ZDC Controller; I had a brief; courteous discussion with the ZDC Controller. It was clear that the Controller was not familiar with the AIM requirement referenced above. I chose to not debate the issue in the air; and Controller said there was no follow-up discussion or action required. What transpired though is reflective of what seems to be a general non-awareness by controllers (and a lot of pilots) of the requirement to follow the hold when radar vectors are not being provided to the IAF.