Flight Instructor with student reported during departure they flew into an area below the minimum vectoring altitude and had a track heading deviation. Reporter stated phraseology used by the controller was confusing and contributed to the deviations.

Narrative

On Day 0; I accepted an IFR clearance to ZZZ1 airport from ZZZ from ZZZ Tower on XXX.X. I was on the ground at the FBO ramp. The clearance was as follows: Aircraft X is cleared to the ZZZ airport via the ZZZZZ X departure; ZZZ2; direct; climb and maintain four thousand; departure frequency XXX.XX and squawk XXXX. Upon taking off I followed the ZZZZZ X departure directions to fly heading XXX to four thousand for vector to assigned route/fix. I was flying on heading XXX at 1300 MSL when ZZZ Tower told me 'Aircraft X; ON COURSE APPROVED; contact Departure.' My student responded with; 'on course approved; over to departure on XXX.XX'. Because I was told 'ON COURSE APPROVED' I started flying towards ZZZ1 on a heading of XXY. ZZZ2 was only 1.5 miles away and it was an initial approach fix for the approach we were going to fly but had not been cleared for. While simultaneously turning towards ZZZ2 on a heading of XXY; I switched frequencies to contact ZZZ Departure on XXX.XX. I am not sure what verbiage I used when I checked in with ZZZ Departure on initial contact. ZZZ Departure responded and said' Aircraft X; ZZZ Departure; radar contact; confirm you are still on the ZZZZZ X departure flying runway heading'. I then responded with 'we are going direct to ZZZ2 as we were given the instruction on course approved'. ZZZ Departure said 'Aircraft X; fly heading XXZ and descend to three thousand'. After a short amount of time estimated to be three minutes; ZZZ Departure said 'Aircraft X fly heading XXA and join the final approach'. At that point we flew heading XXA and joined the VOR Runway XX approach into ZZZ1. At this point; ZZZ Departure informed me of a possible pilot deviation and gave me the ZZZ Tower phone number. I then cancelled my IFR and requested to return back to ZZZ so that I could call the Tower and discuss the possible pilot deviation. I landed and then called the number I was provided. The individual I spoke to was named Person A. They did not provide their last name and I did not ask for it. Person A took down my name and phone number. He then informed me that the possible deviation was because I did not follow the ZZZZZ X departure and started flying heading XXY towards Shafter before I was given instruction to do otherwise. He also informed me that he understood that because I was told 'on course approved' that he wasn't sure whether or not it was the ATC controller's fault or my own. He said that I flew into a minimum vectoring area that had a minimum vectoring altitude of three thousand. He said that because my altitude was below that is why I received the possible pilot deviation. I told him that if I did anything wrong that I take full responsibility for my actions. I asked him what I should do next and he informed me that he could not give guidance on the matter but that he had to report it to the Flight Standards District Office before his shift was over.I have reviewed chapter 5 of the aim and do not see anywhere where the verbiage 'on course approved' is defined. I have also looked in the pilot controller glossary and cannot find anywhere where 'on course approved' is defined. I am unsure if correct pilot/controller verbiage was used. I have only ever heard the phrase 'on course approved' in VFR settings. On google; some individuals define 'on course approved' as an instruction to return back to the previously assigned course usually following a radar vector. If this is true; then my last assigned course would have been the ZZZZZ X departure and I should have remained on runway heading. However; it does say 'return' back to previously assigned course. I do not believe I was ever off course to begin with and I was already flying runway heading. I am unclear what I was being approved for if I was already flying my assigned course. Why was I being approved to fly the course I was already cleared for? Had the controller not said 'on course approved' I would have continued flying runway heading and everything would have been fine. It is my belief that controllers should not say anything that causes confusion or something that is up for interpretation especially when I am already flying the course they want me to fly. I believe the verbiage the controller told me was confusing and vague especially because the ZZZZZ X departure instructions are to fly runway heading until ATC radar vectors me to my next assigned fix. I was flying runway heading until I was given further instruction. I don't understand why I was given further instruction to do what I was already doing. This is why I interpreted 'on course approved' as him telling my to proceed on my course to my next assigned fix which was ZZZ2 VOR. I have talked to several other pilots about this matter. Every single one of them said they would have interpreted the instruction the same as I did and would have flown to their next fix. I even asked two of my close friends who are ATC controllers themselves and they said they would have interpreted the instruction the same way that I did. I had never heard that instruction given under IFR. I thought it similar to the phrase 'resume own navigation' which is defined in the pilot controller glossary. It seems to me that there is some confusion among the entire aviation community about what on course approved means if it doesn't mean resume own navigation to next fix/course. I desperately want to learn what it means but cannot find any truly credible source. I have self reported to the company that I work for on the incident; completed two 'wings' courses; and have searched for a credible source on the phrase 'on course approved' to try and better my knowledge on the matter. I do admit that I should have confirmed that I was cleared direct to ZZZ2 VOR; however; it didn't strike me as an odd or confusing instruction until I was informed that I may had done something wrong. That is my biggest take away from the matter.