Maybe I can help since I own a 2002 Explorer with factory alarm and the 2003 is the same generation. The reason why the factory alarm is not always going off is that you can only set the alarm one of three ways. 1)factory FOB 2)pressing the bottom two buttons on the keypad outside on the drivers door 3)locking the doors while a door is still open.
My bet is that it is arming when you lock the door with the aftermarket FOB and either you or a passenger hasn't closed all the doors yet. Since this might be a rare issue, it seems like the cars factory alarm is intermittly arming.
Personally on my car I didn't even know it had a factory alarm until the above senario must have happened to me and the factory alarm went off. I went running around my house looking for the factory FOB to turn it off, LOL. I had to go back into the install and get my disarm wire off the remote starter and find the elusive PURPLE / green disarm wire in the car.
I experienced the same problem, it is not located in the drivers kick panel near the foot brake and I ended up grabbing it in the drivers door boot. Later on,(I own the car since new)my remotes died and it was cheaper for my to buy a new remote starter with more features then to buy 2 old remotes for the old system. While I was installing the new remote start, I was able to find the disarm wire in the drivers running board. There are 2 bundles of wires running down the running board, it is in one of them. There are a few PURPLE / greens in there, but if you carefully check(and I know you know how to as you clearly stated) you will find it. Make sure you check both harnesses as I don't remember which it was in, and there are a few to check. As stated earlier it is tested while testing for ground while turning the key in the drivers door cylinder.
Also, Negative trunk release was under the drivers sill for the rear seat. I think brown or tan with a stripe. It will test while pressing the button on the rear hatch. And also Negative door trigger with no light delay is light blue/white on the bottom left connector of the cluster. no need to remove cluster, comes straight down on the front bottom edge of the cluster. It is the door ajar wire for the cluster. This wire cannot be used for Dome Light supervision. You still need to use the BLACK/ blue (+) dome light wire for that in the drivers side sill.
Good Luck
Frank