That thickness (or lack thereof) of the cable wouldn't cause the issue you describe. If the cable wasn't thick enough (read: enough copper to carry the current), it'd get warm / hot / melt the insulation. The likely difference in resistance between that and something thicker equally wouldn't cause the voltage drop you're seeing.
Resistance in a cable is a function of it's cross section and what it's made of - the thinner a cable gets if everything else is the same - then the higher it's resistance and the higher the resistance the lower the current could pass - however - given such a short length of cable, I doubt you'd see even a single ohm's worth of resistance difference between whatever gauge that wire was and the next one up.
Just read through this thread (don't know how I'd missed it) and agree it's a great write-up. Yours is the writing style I try to have when I write, except I fail at being articulate and structured. Anyway, I loved this and hope you'll continue with it, so thanks for posting.