Ah. You are correct. Thank you.
Looking carefully at the 3e spell wording, True Seeing confers no specific ability to Detect Magic. It will show things that are altered by magic or hidden by magic. In fact, it is not even a benefit for finding most traps, magical or mundane.
So the answer is "I am interpreting it wrong", as you said.
Haley has a sky high Spot and Search, and found the "trap". Redcloak is relying on True Seeing, but did not bother with Detect Magic, and sees nothing. Detect Magic would presumably reveal something is there that might be worth further investigation. Redcloak has been using True Seeing, and, nobody having been blown up at the foyer in the last 100+ tunnels, saw no reason to alter his tactics.