Who decides the order of finish and records the times?

It depends on the timing system (Red Book C5-313 and C5-308):

• Photo finish (primary): a Chief Photo Finish / E-Timer and an Assistant Photo Finish. The photo finish judge reads the photo and reports the order of finish and times.
• Manual timing (primary or backup): a Chief Timer and Timers, plus a Chief Finish Line Judge and Finish Line Judges.
• Transponders (backup timing and live stream data): an E-Timer.
• Video timing (backup only): an E-Timer, or a Timer under the E-Timer's supervision.

The Chief Finish Line Judge determines the official order of finish when there is no photo finish system, or if the system fails.

Lap officials (required regardless of timing system): the Lap Scorer shows the laps still to skate by the leading skater, turning the lap board each time the leader passes, and rings the bell for the last lap. The Lap Recorder keeps a lap-by-lap written record of the leader's lap times and checks the Lap Scorer (ISU Rule 286).