You are correct! "AS A PILOT"! No showing for a trip isn't documented even if terminated on PRIA. Said terminated employee isn't in the cockpit so he's not a pilot and not documented on PRIA. It's clearly spelled out what is and isn't covered in PRIA. Only actions as a pilot, training and flying. Go pull your PRIA and look. Training and FAA action(s) is what's PRIA designed to show the new employer. Getting fired is a topic for the interview.

Don't paraphrase, this is directly from the FAA below.

PRIA requires that a hiring air carrier under 14 CFR parts 121 and 135, or a hiring air operator under 14 CFR part 125, request, receive, and evaluate certain information concerning a pilot/applicant’s training, experience, qualification, and safety background, before allowing that individual to begin service as a pilot with their company.