In your schematic, option 1) is the correct way to do it. Comparator output > currentlimit resistor > LED anode--LED cathode > ground .
Your schematic should work fine. When power is applied, the capacitor charges very fast through the resistor and the LED should come on. When power is removed the capacitor discharges through the other resistor to ground and when enough charge is drained off the LED should go out. If it works backwards, then your inputs should be swapped around. If the time is too short, use a larger (higher ohm) resistor across the capacitor. The greater this resistance the longer it will take for the charge on the cap to bleed down. Doesn't TAOE use a 6 Megohm resistor here?