Check Airman reported miscommunication with MIA TRACON issuing clearances as incomplete; unclear; and non standard which resulted in an NMAC.

Narrative

I was conducting a Part 135 IFR flight check for a PIC (Pilot In command). We flew a hand flown ILS to minimums at FXE and advised the Tower that we wanted to go missed and fly one more ILS Runway 9(coupled this time). This was all done in VMC conditions with VFR traffic advisories from ATC. Our missed approach instructions from the Tower were to 'make a left turn northwest; climb and maintain 2;000 ft'. Shortly after the missed we were handed off to Miami Approach. On check in with Miami; we stated that we were climbing to 2;000 ft. and on a 290 heading which we continued maintaining VFR. The Controller instructed us to 'turn left to a 270 heading'; a few moments later the controller asked us if we were going to 'turn left to 270' the PIC replied ' yes we turned left to a 270; that is what we are doing' The Controller once more asked us if we were 'going to turn left' the PIC asked 'left to what heading; we are flying 270 as instructed' the Controller replied 'I wanted you to make a left 360 to a 270 heading'. The PIC replied 'that wasn't what you gave us but were turning left'. As we started the turn; I saw what was a very close by smaller jet descending to get out of our way. This must have been the traffic in conflict. Being on a 290 heading and the controller stating 'turn left to 270' we both were under the impression that the Controller wanted us to tighten up our downwind for the ILS. After 3 attempts to turn us 360 degrees to the left; the controller never said the words 'make a 360' or 'make a left orbit' or anything that would indicate to us that he wanted us to turn further than 20 degrees to the left. I do not believe that standard phraseology was used by the Approach Controller and this could have avoided a near collision. If I was under the hood taking a flight check; I would have reacted the same way as the PIC taking this flight check as he was doing exactly as the Controller instructed him to do. I'm not sure why; if he didn't see us doing what he thought he was instructing us to do he did not move the other conflicting traffic clear of us? I am a firm believer of standard phraseology use for communication with ATC and this is precisely the reason why.