Are you measuring 13v between the 'negative' side of the bulb and ground while the light should be working but isn't?
If so, the connection between the negative side of the bulb and ground is faulty.
If the bulb works (and the door switch is in the negative lead) you should read 13v on negative side of the bulb (when the door is closed) as it is not connected to the ground (no current flowing so no voltage dropped across bulb). Just noticed '2E0'. Intermediate isnt it? I don't need to tell you about ohms law!
If the door switch is also used for an alarm (and either the alarm or the light is aftermarket), fit a dedicated switch for lamp.