I drew a sketch. I hope it helps. Assume pigtail is disconnected from socket, as are the wires to the lamps. My voltage drop is occurring "upriver" of where I expect it. I figured on $12\,\mathrm{V}$ all the way to the lamps, then $0\,\mathrm{V}$ after, with both brown and black having to keep the same reading. Looks like I don't understand this.
1 Answer
Very few power sources are "ideal voltage sources." Most degrade when you put a load on them. This is typically modeled by an "internal resistance" of the voltage source that is in series with the source and the load.
When you start to draw a lot of current through a real voltage source, you'll find the voltage goes down due to this effect. How much it goes down depends on your source. A small AA may go down quickly as you pull more current. A 120V wall socket may support quite a lot more current before issues appear (those issues will show up as a "brown out" where the power to the circuit seems to falter)