First Officer reported communications problems led to lack of compliance with established procedures during engine starts on battery power and with a ground power unit that was not approved for use with the aircraft.

Narrative

These statements are true to the best of my recollection of what occurred. Crew was asked to do a maintenance check flight for Aircraft X. This was a ground test. Before starting the engines; we were informed that we would not have GPU available because the battery was dead. We started both engines; 1 and 2 on battery. A full engine run up and test of the bleed valves was completed and all checks were normal. Crew shut down aircraft normally. There was another request to do a second maintenance engine run. This time we had a GPU available to us. Crew started engine 1 with the GPU. At 20% minimum ENG RPM when the Captain introduces fuel with the condition lever; the GPU tripped off and the cockpit had erroneous readings and the engine over temp light was on. First Officer (FO) calls for cutoff and motor. Airplane engine was motored for the limitation time and below 175 degrees C. The crew used the QRH and followed procedures. Maintenance was called and notified. It was later discovered that the GPU used for the attempted engine start; was not our normal GPU and was thought to have come from Company A on the other side of the airport (I don't know officially where the GPU came from). In addition; that GPU is not able to handle a SAAB engine start. The flight crew was unaware of this. The crew spoke to Person A. He asked if we could confirm a light off. The flight crew could not determine of the engine had lit off or not. If we could be certain the engine did not light off; we could attempt a second start. If it was suspected of a light off; then the ETM/FDR (Engine Trend Monitoring/Flight Data Recorder) would have to be pulled for analysis. Person A suggested erring on the side of caution; I agreed. It was suspected that false readings were given due to the GPU being tripped; however could not confirm. I stated to the Captain that I wasn't comfortable starting the engine again; because we weren't certain. While driving home; I received a call from the Captain that I needed to return to the airport because Maintenance wanted to have us do an engine run. I was surprised by this; after what was discussed earlier. I called Person A again and he stated exactly what we spoke about earlier; there were no changes.I told the Captain I was not comfortable with starting on the phone. I returned to the airport and met him on the ramp. I stated again I was not comfortable; he said I was on duty and had to. I told him respectfully I don't and I'm not going to. I exited the airport and returned to the parking lot. The Captain did an engine start on the airplane single pilot. I'm unaware of what happened after. I called Person B to inform him of what occurred. The next morning the Captain stated the ETM/FDR box was being sent out to be analyzed.