B737 Technician reported communications errors and automation difficulties resulted in an aircraft departing in revenue service with an open discrepancy that could not be properly placed on deferral.

Narrative

On the night of DATE; I was responding to a gate call for A/C X parked at gate; ZZZ to ZZZ1; for the following write up created by the pilot; log X at XA:47 'NUMBER 1 ENG FIRE HANDLE DOES NOT ILLUMINATE DURING FIRE TEST'. After initial T/S (Trouble Shooting); I confirmed the fault; and after talking to the lead of [the zone] over the phone; he told me to request an MEL; because the fault was intermittent and could not be cleared. I initiated the request for MEL. While I was in the process of doing so; I accidentally cleared the log page X @ XB:30; instead of deferring it; which was just discovered when I was speaking with M/C (Maintenance Control); over the phone. M/C then told me to open another log and resubmit it. I open the log page Y @ XC:36 'NO 1 ENGINE FIRE HANDLE DOES NOT ILLUMINATE'; but I was not able to submit the MEL; still talking with M/C over the phone asking for help in resubmitting the MEL; M/C said that the aircraft was maintenance released and he could not help any longer. Immediately I called the lead back and asked him if he was the person whom released the aircraft. He replied that the aircraft was released by the Zone Supervisor; maintenance release X @ XD:33. At this point; the crew probably saw the maintenance release been printed and advised the ramp to start pushing the aircraft and they left the gate. I went after the lead to let him know that the aircraft was leaving and I did not have a chance to apply the MEL; because the maintenance release was issued and the aircraft brakes had been released. When I walked into the Supervisor's office the lead was talking to the supervisor for a swap for [the aircraft]; because the deferral for the MEL would take too long; the Supervisor was on the phone; asking for a swap when [they were told] that the aircraft had departure. Supervisor told me to contact M/C and ask for help. I contacted M/C but nothing could be done while aircraft was in the air. Aircraft X landed safely at ZZZ1 and log Y was cleared @ XE:50. This is how I remember this incident to the best of my knowledge.