Because the "ground" is being interrupted by the door, the Pink wire isn't really the "hot"
wire--I know that is probably confusing, but ask the question, "what makes 'hot' hot?"
Unfortunately, there are two answers to that question:
1. The pink wire is "hot" because it is at 12 Volts relative to the chassis--which acts as a "ground" return.
2. The pink wire is not "hot", because the hot side is the side connected to the switch. In fact, one could argue, at least for this one circuit, it's a positive ground system--in other words, the Pink wire is ground, and the chassis is at -12 volts (i.e. it's a "hot chassis").
So, think of it in terms of current loops. Look for the completion paths. The pink wire is connected to the positive side of the battery. The switch is connected to the negative side. When the switch closes, it completes the path from the positive side of the battery, through the dome light and back to the negative side of the battery via the door switch.
If the LED strings are connected across the dome light, then they will be energized in the same way the dome light is: when one of the car doors is opened, the dome light and anything connected across it will come on. When all of the doors are closed, the dome light, and anything connected across it, will turn off.
The only role the cigarette lighter plays is to supply a ground connection. Nothing else. Assuming that is a consistent connection to the negative side of the battery, that is all it will supply -- a connection to the negative side of the battery, no matter what state the cigarette lighter is in. If it doesn't, then it isn't the kind of connection you want.
So, now, that leaves the switch. I'm struggling to understand what the switch is for. You say you want "that one change in the truth table" but I'm not sure what change you are talking about. You, also, say, "I can’t imagine a time when I wouldn’t want the lights to come on when opening the door." That will happen if you hook the LEDs across the dome light,
But, you also say, "I would rather be able to leave the switch in the on position and the lights not be on all the time" -- so, when _do_ you want the lights to come on -- I thought, when the door is opened? And then you say, "this way when I turn on the car, the lights come on automatically without my flipping a switch" So, I still need to know when you want these lights to come on, because that makes it sound like you want them on whenever the car is "on" (or did you wanted the circuit to be "active" (i.e. with the potential to be turned on)).
Can you give me some kind of "truth table" with the three conditions: car on or off, door open or closed, switch turned on or off. There will be 8 lines (or conditions) in such a table, and it would really help to know what each of those conditions are. You can do it like this:
Switch off, Car off, Door closed: LEDs ???
Switch off, Car off, Door opened: LEDs ???
Switch off, Car on, Door closed: LEDs ???
Switch off, Car on, Door opened: LEDs ???
Switch on, Car off, Door closed: LEDs ???
Switch on, Car off, Door opened: LEDs ???
Switch on, Car on, Door closed: LEDs ???
Switch on, Car on, Door opened: LEDs ???