ATP Flight Instructor with student reported a CFTT event during climbout from a low approach due to a miscommunication between Pilot and ATC. Reporter stated he did not confirm the alternate departure instructions resulting in the CFTT.

Narrative

Conditions were VMC clear below 12000 and visibility better than 20 miles. My student and I were cleared for approach into ZZZ. The controller requested our intentions and I stated this approach would terminate with a low approach only and runway heading then direct to destination when able. The controller acknowledged our intentions and shortly after we were authorized to change to advisory frequency.During our climbout; we checked back in with Approach and it did not sound like the same controller. He stated the missed approach procedure was a right turn. I replied we did not request the missed and were runway heading climbing. He acknowledged my reply then shortly after asked if I could maintain terrain separation until 5200. I replied affirmatively. We were then given direct destination and climb maintain 6000. After some time of being level at 6000 I was contacted regarding the possible pilot deviation.I feel that the alternate missed instructions were not confirmed by me or the controller(s) even though my intent was clear before changing to advisory that we had no intention of flying the missed procedure. This lack of clarity and possibly our intent not being relayed to the next controller; resulted in our left turn away from the terrain in question being instructed and complied with in a timely manner. We remained several miles away from any terrain and at no time was there risk to safety of flight.