OK Trap sense says-
Trap Spotter (Ex): Whenever a rogue with this talent comes within 10 feet of a trap, she receives an immediate Perception skill check to notice the trap. This check should be made in secret by the GM.
The section on covered pits says-
Covered pits are much more dangerous. They can be detected with a DC 20 Perception check, but only if the character is taking the time to carefully examine the area before walking across it. A character who fails to detect a covered pit is still entitled to a DC 20 Reflex save to avoid falling into it. If she was running or moving recklessly at the time, however, she gets no saving throw and falls automatically.
So a rogue is running towards a covered pit, normally no save, the rogue however has trap sense so by RAW can detect it 10' away. How would you adjudicate this?
My inclination would be the rogue gets the DC20 check at 10' away and if successful detects it and can just stop, 10' is a long distance to be able to stop in.
To be clear the question is about trap spotter not trap sense (which gives a bonus on saves and AC versus traps).
Beyond that however everything looks correct. If the rogue didn't detect it and stop in time and wasn't being reckless they would get their trap sense bonus on the save throw to avoid the trap (provided they had trap sense).