A Center Controller reported an IFR aircraft departed a non towered airport despite being assigned a hold for release and did not comply with the published departure procedure resulting in flight below the Minimum IFR Altitude.

Narrative

Aircraft X called from the ground at ZZZ requesting clearance to ZZZ1 at 8;000 ft. I had an arrival and departure and ZZZ1 so I advised Aircraft X that he would be a hold for release clearance. I read the clearance and Aircraft X said that my transmission had been broken and gave a partial readback. I restated his altitude and told him to monitor this frequency for his clearance. He read back 9;000 ft. and to monitor my frequency but didn't read back hold for release. Immediately afterward I got a call from Sector X about a ZZZ2 arrival that had traffic in the ZZZ VOR area and my attention was diverted. When I got the call about the arrival at ZZZ1 and got the departure on radar I attempted to call Aircraft X back and issue his release for departure. Aircraft X did not respond.I responded to several other calls in between reattempts to contact Aircraft X. Eventually I observed Aircraft X on radar. He had already departed and was heading north. When he called I identified him and issued him 8;000 ft. He continued to head north toward 8;900 ft. MIA area and I asked if he was doing the published departure procedure. Aircraft X responded 'published departure'. He continued northbound and was encroaching on the 8;900 ft. area at 7;500 ft. I again asked him to verify he was doing the published departure procedure and issued him a climb to 9;000 ft. The controller who was relieving me verified in the ERIDS that the published departure was not what Aircraft X appeared to be doing. Aircraft X said he wanted direct ZZZZZ and I cleared him to ZZZZZ reaching 9;000 ft. Use unambiguous phraseology; make sure pilots and controllers are familiar with departure procedures; have enough staffing to split sectors and staff D-sides when necessary