If I were to place a bet, I would put it on motherboard caps. The IIcx/IIci are of the age that its needed, period. I have several and they all need a re-cap. At the very least wash the board well with hot water and light amount of dish soap. Paintbrush works well without being too aggressive. Dry with compressed air or hair dryer. I use a hairdryer and set them in front of a space heater. Then occasionally take the dryer to it. A lot of water can sit in the PDS slot holes and under surface mount chips. Its vital its 100% dry. Cleaning may get it to boot since cap goo is conductive (and corrosive). If it still does not work, the caps are too far gone or they corroded some traces. I use vinegar and a tooth brush to clean traces and remove green corrosion (1/2 hour max soak time, wash again with water/soap).

You may be getting different responses from different power supplies due to output. With bad caps a subtle difference can be a big deal. You can also leave it plugged in for about 10 minutes then try booting again. As mentioned, use the keyboard too. Both power-up methods use a slightly different startup path in the circuit.