Flight Instructor reported while flying a VOR approach they descended below the published altitude resulting in flying towards terrain. They recognized their mistake and executed a missed approach.

Narrative

My client; the pilot flying; and I; the flight instructor; were practicing the VOR-B approach into ZZZ. The conditions were VMC and we began the approach at the ZZZ IAF at 3;000 MSL and were cleared for the approach at some point on the initial approach segment. My client began a descent down to 2;300; the minimum altitude on that segment of the procedure; and ZZZ [Approach] gave us permission to switch over to the common traffic advisory frequency somewhere outside of ZZZZZ; the intermediate fix. Upon crossing ZZZZZ; my client began a descent to MDA. Neither of us realized at first that we had begun a descent prior to the FAF; thinking that ZZZZZ was instead the final approach fix. As I preceded to make a radio call to local traffic I noticed the distance change on the GPS and realized that we had actually just crossed the final approach fix several hundred feet below the minimum altitude. The weather conditions were VMC and we could see that there were no obstacles so we levelled off and then executed the missed approach procedure. Upon contacting ATC on the missed approach; the controller advised us that he had gotten an altitude alert and that we had to maintain 2;300 MSL until crossing the FAF. We agreed and acknowledged that we were aware and would not let that happen again. At first I was not exactly sure how either of us had missed the mistake. I've been flying professionally for many years and had flown for the airline as a First Officer and Captain for most of those years. I put strong emphasis on a sterile cockpit and there we no unusual circumstances or distractions during the event. In the debriefing we discussed the possibilities as to what had caused the error and it may have been an instance of confirmation bias. Prior to the flight we had briefed performing the procedure turn at ZZZZZ; which also served as an IAF; and I noted that ATC rarely assigned ZZZ as the IAF as I had done the approach many times before. As result; we were not expecting ZZZ to be our assigned transition onto the approach; and even though we briefed the approach again in the air; prior to beginning the approach; and noted that the procedure turn at ZZZZZ would not be required since we were starting at ZZZ; we still managed to confuse ZZZZZ for the FAF. The other mistake we made was not monitoring the Approach Controller's frequency after we were advised to change frequencies. Normally I make it a habit to monitor the Approach Controller on the second radio after switching to CTAF. Had we done that we would have heard the controller issue an altitude alert and could have caught the mistake sooner. For future flights; we agreed to maintain better altitude awareness because such a mistake in IMC could be fatal.