No that is not normal. When your car is locked with the fob the anti-theft alarm is set. It seems that the alarm system (when you open the door with the key) recognizes the door opened, BUT does not recognizes you did it with your key. This is just a guess, but I would use some Deoxit5 (contact cleaner) or some Houdini (lock lubricant) and spray it through the key hole then insert your key and turn it several time. MAYBE it will free up a switch inside the door and then the alarm will recognize you used your key vice the alarm only seeing the door opened.

If that doesn't work, hopefully a scan from a good tool will give you a direction to follow.