Sounds like something is shorting somewhere alright, but I’ll wager that the “thin strip of wire” you’re seeing between those pins of the LCD header is actually a 4.7Kohm resistor (you can check this with a multimeter):
-(thanks for the pic Jim!)
This is a modification made to the later (black bezel LCD screen) to keep the newer screens from interfering with the programmer. If you’re interested you can read all about it in this thread
Actually, do you have a multimeter around? You could use a continuity checker to see if there is a short between Vcc. It might not even be a total short, you could have a low resistance connection, or a backwards diode. Anyway, whatever the problem, its clear that somethings amiss.
-Adam