Chosen Solution
Can I replace this socket, resolder it, or is it held on enough by the bottom connectors (which are intact) that it will work fine?
The solder pads have been ripped off from the anchor pads that used to be secured to the board. What I would do is use either epoxy or superglue carefully to adhere the connector back down flat to the board. If you do go this route make sure to not get any of it in the connector. You can use a bit of tape to hold it down when waiting for the glue to hold down.
Best solution is just buy a new Backlight Board $15 on ebay. This board is where the VSYNC connector is located, VSYNC connector is NOT on the Logic bd. (Apple iMac mid-2011 A1312 27" LED Backlight Inverter Board 612-0094).