[BrianG]

Yeah, that's a "switched ground" setup and is pretty typical. Since the LED strips aren't drawing much current, you should be able to get away with using the door switch as the ground without any added components. So, it would go like this:

+12v from the vehicle wired to the LED's positive input

Wire the LED's negative input to the yellow/green wire.

That's it.

The confusion probably stems from the fact that almost every circuit you see has the switch between the positive supply and the load. But really, it doesn't matter where the switch goes in most circuits as long as the circuit it broken.

EDIT: Even though both of those diagrams say the wire is yellow/green, I would check to make sure! Simply attach a meter's ground probe to that wire, and the meter's red probe to a 12v source. If that wire is correct, the meter should show 12v when the door is open and 0v when it is closed.