Spanners and NSEU,
Disregard the inlet door 15deg position. That's for AIR mode.
My take on this is..... Having observed the 744 APU inlet door open/close from the ground, it takes about 30secs max (if that) either way. The 13deg sw is telling the APUC that the inlet door is indeed open and is on it's way to the full open 45deg position as confirmed by the actuator internal limit sw. Needless to say the APU start sequence is well on it's way inside 60secs, but ONLY after the APUC has seen the actuator internal limit sw signal.
McHale.